This is basically not a guide. There is already a great Karma guide I don't want to compete with nor I would stand a chance if I did: Egypsian Lover's guide. It covers nearly everything you want to know about Karma and it suggests one way of building with tons of situational adjustments. I very much recommend reading that guide to everyone, even to guys who are not interested in playing Karma themselves.
However that guide is mainly based on older experiences and does only adjust to changes instead of completely rethinking the build. So I suggested the build shown here as an alternative. I first suggested it in the comments and both Egypsian Lover and me wanted to get further play testing on it. Egypsian Lover does not have much time and we didn't reach anyone else open for testing by the comments section so I decided to get it into an own build here with reference to his guide. Here we go...

This section in this guide is NOT meant to show you when to use which ability, or what advanced tricks exist with Karma and her abilities. Nor shall it show you how to make your abilities hit. For all that I strongly recommend checking Egypsian Lover's guide, it is perfectly explained and illustrated there, I do not want/need to redo work that is already done.

This section IS meant to show experienced Karma players in what style I play her (very offensive first line support), why I do that and what stats requirements arise from that so I can afterwards show how I get those stats. So lets go:

Laning

I play her duo lane as support. I pick only a minimum of farm so I need a cheap build, while giving my carry mate the chance to get a very expensive build quite early.
I play Karma very aggressively, jumping out of the brush into my lane opponents hitting Mantra Soul Shield + Heavenly Wave. If my opponents retreat I get off some auto attacks, if they go for me I retreat back into the brush and further back. I use Spirit Bond only if i see a significant advantage, otherwise I save the mana.
This combination usually provides a positive damage trade even if my carry doesn't join my move (thanks to the shield effect and burst damage). If my opponents now chase me into and through the brush they might turn the trade, but this also gives my mate time to get off damage. I repeat that even if I am injured, as the missing life only increases my AP.
When seriously wounded I use Mantra Heavenly Wave to heal me, always trying to hit either my mate or my opponents as well.

So far it sounds easy, now lets have a look at the difficulties:

Jumping out of the brush requires a brush. Thus it is essential to have a proper fighting spot. You don't want to be pushed into your turret, nor do you want to push the wave into their turret. Thus push control is quite important.

You need to stand some damage, some tankiness is required.

All abilities deal area of effect damage. Thus you always risk hitting minions with it and thus pushing the lane. as mentioned before I don't want to push so I have to keep track of the minions and how much I can risk damaging them.

I don't want to be without a Mantra charge, as it is very helpful for emergency. Though I also don't want to have 2 Mantra charges up, as it is basically a waste of its cooldown. Thus I try to use Mantra whenever I get the second charge: either I do an aggression as described above or I heal me and my mate.

Also early mana is an issue, until you get your Chalice of Harmony.

Thus there are quite many things that require attention, which needs to be practised. Also a carry knowing to control push is required.

Baiting

Karma is great when low hp due to her Inner Flame. Using Soul Shield and Mantra Heavenly Wave + Heal/ Barrier makes you surprisingly hard to kill. Also the Heavenly Wave damages a bit and at least flags opponents for kills/assists. Using that combination under a turret when low will get you a kill every now and then, learn to know how far you can go (you can go very far), your opponents do not know that.

Jungling

I do not suggest playing Karma as a jungler and although it is possible I do not want to deal with that here.
What I want to do here is talking about taking jungle creep if you get the opportunity: avoid it. Usually the jungle creeps are for your jungler and you do not need the farm. If you feel you need the farm you might take ghosts or wolves but usually you enter jungle only to place wards or help your teammates or possibly for escapes.
Blue Buff is sometimes offered to support. As both cooldown reduction and manareg are handled soon by something else you don't need it. Early on it helps a bit but isn't necessary. Your ap carry or jungler should benefit more. So if an early blue buff is offered you may take it but later in the game you should only take it if otherwise your opponents would get it.

Teamfights

In mid and end game there is not much to do alone for Karma. So it is basically about team fights. In team fights there are two major questions to answer: Whom shall I support and where shall I position myself. So look at our skills:

Soul Shield

prevents a given amount of damage. That amount varies with AP and therefor with your hp percentage. It does not depend on your target, so you should focus on targeting your carries (both ap and ad) with it as their hitpoints are worth most. Of course if someone would die without the Soul Shield or you need the Mantra effect that general rule changes but in most cases carries are the targets.

conclusion: be in range of your carry

Mantra Heavenly Wave

is area of effect and both damages enemies and heals mates by a percentage of their missing health. Of course if someone has more total hp there is more potential missing hp and therefor more potential effect of Mantra Heavenly Wave on them. So there are many parameters affecting the efficiency of its use: Number of opponents hit, their hitpoint's value (damage carry or tank?), the number of mates hit, their absolute missing hitpoints and the worth of their hitpoints. These are too many parameters to calculate the perfect placement of the rather short range cone shaped spell. But we can derive the following:
This spell mechanic favors healing tanky champs more than any other heal or shield spell in the game. It also favors targeting areas where as many champs (of both teams) as possible are and that is the center of the fight, also usually where the (off-)tanks are. Heavenly Wave has a rather short range so:

conclusion: be near the center of the fight.

Spirit Bond

Is relevant for the guy that starts fleeing, either on their team or on yours. Usually that one is your or your opponents carry. Be flexible with that spell.

conclusion: stay in range of both your and your opponents carries. -> center of the fight.

Inner Flame

If you drop low you get stronger.

conclusion: do accept some damage (for a good trade) -> be in the fight

Combined positioning and skill directions

overall you should be somewhere slightly behind the center of the teamfight while using
Mantra Heavenly Wave in forward direction towards your tanks and opponents carries
Soul Shield in backwards direction to your carries.
Spirit Bond in the direction it is needed most.

Note: these are guidelines showing what I do mostly, however some less frequent situations require the exact opposite usage of the spells. Experience is required here.

Stats requirement conclusion

Early game we only need some tankiness against the opponents ad carry which leads us to the use of flat armor seals. Other runes are used for later game and are thus scaling.
Later on we need some AP to show effect and quite much survivability to be able to position correctly to apply our spells effective. For that we will get defensive summoner spells, some def and ap/def items.
Of course we need cd reduction like any support.
We need some sort of mana engine. I prefer a regeneration based mana engine as we got life sustain by default (healing) and then can regenerate in all ways staying in defense while others got to recall.

There are a few summoner spells to be concidered:
Exhaust Flash Heal Clairvoyance and Barrier

Clairvoyance is cool if properly used, but late game Twin Shadows does the same job and early game warding with Sightstone does. This one is a matter of preference.

Exhaust has a slight and often unnoticed problem: its range is shorter than most ad carrys ranges. Thus while fleeing you have to turn to apply it what you actually do not want. Additionally it is less surprising than the others as it takes 3 seconds to show its full effect. I do not recommend it for Karma as baiting is actually a combination of fleeing and surprise and thats her strength.

Heal returns a good amount of health to your team, helps with baiting, increases your hp by the mastery ... it is basically made for Karma. I used it in season 2, now it is even better.

Flash gets you out of situations where no other summoner spell helps but it has a rather long cool down. When used during a bait it often makes opponents turn which might not be your intention.

Barrier helps to survive at low hp and thus profit more from Inner Flame. It helps getting out of many situations if well timed, for example activate when the killing blow is already traveling often lets the opponent turn already and thus shows the same effect as Flash. Barrier helps in many situations the same way Flash does but not in all. It also helps in situations where Flash doesn't help and its on shorter cooldown.

I prefer Barrier and Heal but the others aren't bad either.

In Season 2 I used defensive masteries for Karma but they now removed the cd reduction from that and made utility much better. So I returned to the obvious straight forward utility mastery build. There is nothing surprising in this build and for those choices you might have picked different it doesn't really matter. Not much to say here.

Marks:

Greater Mark of Magic Penetration: Best increase in damage we can get. Damage isn't our main purpose but we deal quite some and we can't get anything else out of (primary) marks. I personally do not like secondary runes so I use them and have not tried anything else. You might try:
Greater Mark of Scaling Ability Power: this favours heal and shielding power over damage, but damage is a part of what you want to do so I think magic penetration is better.
any other useful secondary mark ...

Seals:

Greater Seal of Armor
Those runes provide the early tankiness to survive offensive moves during laning. Also our item build contains magic resistance as core but no armor so those runes fill that gap aswell. This should be the primary choice.
Greater Seal of Scaling Armor
I think flat is better but scaling might also be an option. Maybe also an appropriate mix.

Glyphs:

Greater Glyph of Scaling Cooldown Reduction
We get 10% cooldown reduction from masteries 15% from boots and those glyphs and the quints provide the rest to achieve the 40% cap at level 18. Relying on cooldown reduction items would heavily impact the flexibility of our item build thus I recommend those runes.
Other options would be:
Greater Glyph of Scaling Ability Power: AP is important and those runes are good, just don't forget to complete your cooldown reduction somehow.

Quintessences

Greater Quintessence of Scaling Cooldown Reduction
The Glyphs alone are not enough to reach the cap. If you chose to get your cooldown reduction another way and thus picked other Glyphs you should pick other Quints aswell. Try:
Greater Quintessence of Ability Power
anything you consider useful

Starting item: Faerie Charm + 2 * Sight Ward + 2 * Mana Potion

Yes I do not spend all my gold initially. This is because: Usually I do not need more consumables before getting my Sightstone and Chalice of Harmony and I do not need any other cheap enough item. Still it might be adjusted to setup:
If your opponents jungler uses stealth you might favor pink wards.
If you expect a hard lane you might want another Mana Potion.

Core build

Your core build is Kage's Lucky Pick Sightstone Boots Chalice of Harmony Ionian Boots of Lucidity Ruby Sightstone and in most cases that's the order of purchase. However feel free to adjust:
If you take alot of magic damage (from their support) get Chalice of Harmony earlier.
If you are out of wards and you can't afford both Kage's Lucky Pick and Sightstone take the Sightstone.
If you run around alot, still have a Mana Potion and do not need magic resist pick Ionian Boots of Lucidity before Chalice of Harmony
Add a Chain Vest if you are really really troubled by physical damage.
If you need the third ward earlier get Ruby Sightstone earlier.

Situational building

Ask yourself what statement fits best:
You want more mobility: Enchantment: Alacrity basically this is not a question if you buy it just when to buy it.
You got tons of gold: Rod of Ages (you should be able to accomplish it up to ~27 minutes into the game).
You fear AD damage: get Chain Vest, Zhonya's Hourglass
You fear AP damage: get Mikael's Blessing
You fear all of them a bit: Zhonya's Hourglass
Your team fears cc: Mikael's Blessing
You don't fear anything and your AP carry is fed: Will of the Ancients
You don't fear anything but your AP carry is avarage: Rabadon's Deathcap (in fact this is the choice quite often)

After getting the one of those you need most you should again pick out of these options except Rod of Ages. The one you have already picked might be replaced the following way:

got Zhonya's Hourglass and still scared of AD: Randuin's Omen, Thornmail

got Mikael's Blessing and still scared of AP: Twin Shadows or Abyssal Mask

got Rabadon's Deathcap and still nothing to fear: you are most probably winning. Consider Will of the Ancients, Lich Bane, Zhonya's Hourglass or Aegis of the Legion

All over the game

:
get Oracle's Elixir if necessary and you expect to be able to hold it for a while.

Items I did not pick

Philosopher's Stone Eleisa's Miracle

gp5 Mana regeneration and stonger Heal would be nice and it doesn't even consume an inventory slot. But the hp reg is not strong on Karma as she already has a build in heal, always healing herself when used. Also she often prefers to be slightly damaged over full hp because of her Inner Flame. So all that hitpoint reg is quite bad and its a major part of Philosopher's Stone. Still you would have to add another mana regeneration source to get enough mana while Chalice of Harmony totally does it alone.

Frozen Heart

has cooldown reduction. You want to have your cooldown reduction early so you need to get it into core, where again it is too expensive to get you to a flexible point early.

Shard of True Ice

It provides manareg ap gp5 and a useful active which sounds quite good. However it doesn't provide enough manareg thus you would have to add something else into core getting your flexibility delayed which you do not want.

Tear of Godness Archangel's Staff Seraph's Embrace

As an alternative to Chalice of Harmony it could actually be quite good. I simply haven't had the time to test that yet. Maybe I should do.

Aegis of Legions Runic Bulwark

It has some hp reg that I do not like. It leans towards magic resist which I already have in my build from other sources that are hard to replace. It is often built by your tank already. But in an appropriate situation it still could be a good pick.

Enchantment: Captain

In my eyes it is an enchantment for initiators like Amumu to help the team enter the fight and for back line supports such as Janna to help the team retreat. Karma is in the center of the fight where she needs her own mobility.

others

think about it for a moment, if you do not figure it out just ask in the comments section or check Egypsian Lover's guide.

This chapter is for completeness only and makes this build stand alone although I recommend reading Egypsian Lover's guide as this one is meant as an extention / alternative build path to his guide. For Skill Sequence he suggests picking very situationally, I broke it up a bit (i pick Spirit Bond earlier than he suggests and do not start with Heavenly Wave :
Soul Shield as first, because your Heal gets strong only when the healed target dropped low. And until that happened you usually hit level 2.
Heavenly Wave as second to heal harass that you did not manage to shield except your jungler is just approaching for a gank. Then you might consider Spirit Bond -> Heavenly Wave at 3.

level 3:
If I feel confident I pick Spirit Bond.
If we dropped low in an early bad move I pick Heavenly Wave.
Otherwise Soul Shield

level 4:
Spirit Bond if i do not have it yet, except we are pressed to our Tower and do not move at all.
otherwise whatever is open for picking.

level 5:
Level 2 Soul Shield Level 2 Heavenly Wave Level 1 Spirit Bond is what you should have now. If you don't think you need Spirit Bond yet you may further skip it but get it before end of laning phase.

level 6 - near end of laning:
Heavenly Wave and Soul Shield equally if you don't get harassed and do use all your power to harass yourself.
Otherwise Soul Shield as much as possible Heavenly Wave level 3 if you get harassed hard and can't handle it by shielding.

up to 18:
Soul Shield first Heavenly Wave second and Spirit Bond last or what ever you feel best with.

general synergies

Mantra Heavenly Wave heals more on targets with more health (and thus more potential missing health). Thus she is good at supporting tanky champs and is generally good an tanky setups. Additionally:

Mantra Heavenly Wave heals more when the target is low. Thus Karma can keep a low mate alive for quite long. That is why she synergizes with champions that profit from being low in some way:
Poppy for her Valiant Fighter
Olaf for his Berserker Rage
Irelia for her Equilibrium Strike
Dr. Mundo for his Blunt Force Trauma
maybe I've forgot someone, don't know.

Karma synergizes with Singed and Volibear because:
Spirit Bond helps them reach a target to Fling/ Thundering Smash and the flinged target will automatically land within Spirit Bonds beam and thus be damaged and slowed right after.

common lane mates

as described in the play style section Karmas laning is about burst combos. There are two ways for a lane mate to help with that:
Burst aswell and
assist retreat with cc.

Thus good mates are:
Graves Vayne for their burst and cc.
Caitlyn Ashe Kog'Maw Draven for their crowd control.

playable mates are those who neither burst nor cc stong enough, but instead rely on poking and their own mobility or survival moves such as:
Corki Ezreal Miss Fortune Sivir Varus

a horrible mate is:
any player that pushes like hell, independent of the champ.
Tristana as she always pushes like hell.

uncommon lane mates

As uncommon lane mates are uncommon I don't have too much experience with them. Though I have at least some experience and it was quite good, so I think it should be mentioned here. But still some things said in this part are speculative and deduced from theory, especially when talking about double support lanes.

Karma is a very flexible champ and works quite well with many uncommon lane mates that usually do not go for a duo lane. Knowing that allows to switch your toplane and ad carry in case your toplane got counter picked and helps to get the best out of an accidential random pick.
Still playstyle has to be adjusted a bit.
As I gonna talk about horribly many champs I try to categorize them and only name examples but I won't list all champs.

Assassins (surprisingly strong)

Assassins are those squishy champs with a gap close ability and significant burst damage, such as Fizz. Unless fed they can't burst down an opponent alone, and are quite fragile so they risk dying when they don't kill their target. However their burst + a Mantra Soul Shield gets quite dangerous for a squishy ad carry and the shield effect helps them survive even if they do not kill. So when laning a Karma + assassin lane its up to the assassin to initiate (for the greater surprise effect) and Karma "only" assists with a Mantra Soul Shield and Heavenly Wave.
Many assassins get behind their targets, so a Spirit Bond on your mate will drag it over the opponent and automatically apply its damage and slow (any damage counts in that case).

tanky assassins (quite playable)

Those are guys quite similar to the above, but they are more tanky and less bursty, so they can (and have to) jump in more often for a kill. The playstyle as also the same as above, its simply not that strong, though still playable.
Especially Poppy should be named here due to her general synergy with Karma mentioned above she should actually be a surprisingly good mate.

sustained bruiser with poke (quite playable)

These are guys such as Gragas or Gangplank, who usually go top lane and slowly drop their opponents health while regenerating their own. After a while the opponent either recalls or dies to stupidity or a gank. When laning with a Karma they basically do the same against 2 opponents. Karma simply shields and heals to increase their sustain and assists poking by throwing in Heavenly Wave from relative savety. When an opponent is poked low Karma can significantly help finishing him off with her burst combo.

sustained bruiser without poke (weak)

Here we are talking about champs such as Nasus. They usually simply farm for getting stong late. They can't stand that in a duo lane. Even if they get their farm Karma will be gold starved as she won't get any assists, and in fact Karma needs some assists if she want's to keep up.

poking bruiser without sustain (weak)

Champs like Jarvan IV can't sustain themselves and I don't think Karma can manage that alone, though I'm not sure about it. This requires testing.

Tanks (weak)

Guys such as Rammus, who stand tons of damage and provide significant crowd control but do not deal much damage. If Karma had unlimited mana she could poke and kill the opponents while being protected by a tank. In fact she will simply run out of mana before getting a kill and then the opponents are free to do whatever they want to.

Supports

Supports can't be handled all at once as they are very different from each other. And as it is nearly never seen I don't have enough experience to say much, I can just speculate a bit:
I guess most support + Karma lanes will run into serious mana troubles before achieving anything. An exception to that might be Soraka + Karma, which might turn out to be quite playable. Their combined support should get them out of nearly everything, while Karma can do her offensive moves often enough to set pressure on the opponents. Say thanks to Soraka the mana battery.

As any champ Karma hates some opponent champs and likes some opponent champs, mainly in laning:

Karma hates

Soraka simply for her general early game strengh. A well played Soraka hinders Karmas early game so Karma doesn't get any assists and thus gets gold starved.
Alistar because his Headbutt Pulverize combo hits all its damage before you can hit your Mantra Soul Shield so you simply waste the Shielding part and do rarely hit both opponents with your damage. Also his tower diving power even overcomes Karmas baiting power.
Shacos early strength can shut down botlane which makes Karma gold starved. Additionally Karma always fights while running away giving Shaco free Backstabs. And finally his tower diving might overcome Karmas baiting.
Veigar as his Primordial Burst scales with your ap and thus with your Inner Flame

Karma likes fighting

Blitzcrank as you just don't have to run in on your own for your offensive move when you get grabbed by Rocket Grab. He basically outplays himself or simply does nothing in lane. (Just don't push).
Zilean as nothing is easier than shielding Time Bomb and you can heal it if you fail to shield. Also the surprising burst of Karma sometimes makes Zilean fail his Chronoshift
Tristana as her auto push always provides a nice fighting spot in lane.

I use top posting here:
9.1.2013
Reviewed and updated all of the guide to season 3 changes. It took me a while to test and get a proper build as many core stuff has been removed and my build required significant changes.

28.11.2012
Added an alternative starting itemization in case you really need a ward.
Added Mates and Opponents Section

28.9.2012
I'm back after a period of not playing. They nerved Doran's Ring meanwhile, that's bad but doesn't change the build.

9.2.2012
added Rod of Ages and Locket of the Iron Solari as optional items.

13.1.2012
added 0/21/9 as alternative path for mana hungry players. Thanks to Kittykatman.

14.12.2011
Todays patch reduced mastery improvement for Heal to 10%. Removed the sentence about that in Surge vs Heal comparison as it now doesn't make sense. no other chances necessary due to that patch.

11.12.2011
added play style section
added second char sheet showing some of the options my build contains (especially CV and improved CV/Flash mastery)
added "items i did not pick" to items discussion
added Thornmail to optional anti ad items.

10.12.2011
removed some unnecessary wards out of the item build summary
added this change log

9.12.2011
published original version