This guide is made by a guy who is pretty damn good with Karma in 5v5, but isn't a pro, per se, in LoL quite yet. Adhere to these guidelines loosely, as I know for a fact there are better AP builds out there.

As you glance over the build on this Karma, you will notice she has quite a few AP boosters and CDR. This is because Karma essentially thrives on CDR and AP boosting. Her passive helps with that, but is less of a helper than it is a peripheral to her overall talents, so to speak. Following the below guide will help you not only maintain an AP Karma, but also control her quite well, and make her less of a "hard to use" champion.

To start off, you want to use the two following Summoner Spells for this build: Flash, and Clarity/Heal (depending on how well you can boost her quick enough with AP; if you boost her quickly, use Clarity to keep your mana pool alive)

Flash - Use Flash here for the same reason you'd want to use it on anyone: To gank from a hiding spot, to escape by flashing over a jungle barricade, or to just get closer/farther away from a champion/turret. In instances where you mantra-charge your Soul Shield, ganking will enhance Flash's usage so that you can fly in towards the enemy, mantra-shield on them, then spirit bonding them so they can't quickly escape. Having a slow-down buffer from jungling will further enhance this technique's efficiency.

Clarity(Heal) - Clarity will keep your mana pool alive when you need it most. In extended battles (especially in early game situations) your mana will drain faster than you'd like. Keeping this in mind, Karma can't do ANYTHING without mana. I know you may be saying at your monitors right now "Well, duh! No one can do anything without mana!" and to that I say this: Karma's base attack damage: not that great at all. Not even worth boosting with items or consumables. Thus, keeping your mana levels high will let you stay in group fights longer, and help you fight back a harassment easier. If you think your mana will be fine, or aren't very great with Karma yet, I suggest otherwise to use Heal.

You want to gold/xp farm whenever your team is on the offensive, and the lanes are just about evened out with champs (such as two assassin-types against a huge tank). Using a combination of spirit-bonding the last minion in a line of them, then running the bond through other minions, followed by an immediate heavenly wave can clear minions rather fast considering your AP boost. Spirit-bond damaging minions is a CRUCIAL point in farming with Karma, as you can wound as many as 20 minions at a time with just one cast, if used wisely. Refrain from mantra-shielding here, as you will need mantra and shield use for champion pressuring, especially when the enemy catches wind of an area of their some 20 minions on your turret, and your icon in the middle of them making them disappear in seconds.

In jungling, I found that the golems and specters are very susceptible to mantra-shield and then an immediate heavenly wave. Also, keep in mind that the buff holders will be strong against your attacks, so don't go in there alone, or with less than full hp. If you can persuade a teammate to assist you in killing/backing off before he dies, the blue buff will help you greatly in battle as it lowers your cooldowns.

For this build, it is VERY IMPORTANT that you level up/max out your soul shield first! This will be your savior in mid- to late-game pressuring, and also a great method in gank-saving a pressured ally. Afterwards, you should level up Heavenly Wave, and finally, Spirit Bond.
Keeping in mind the basic fundamentals of leveling spells, you should have each spell at level 1 by champion level 3, using the above leveling order. By level 10, you should have maxed out Soul Shield, only breaking to level up Heavenly Wave and Spirit Bond in limited amounts(They don't allow constant single-spell leveling). Finally, max out your Spirit Bond, so late-game your enemies will be confused how slow you make them, and how much damage the bond does to adjacent champs.

As you may already know by reading an article about Karma, Karma is the only champion who has only 3 levelable spells. Her ultimate, Mantra, is available automatically from the start of the match, and requires no leveling. It acts as a boost to enhance other spells, which each consequently have a maximum of level 6.
Taking this time to talk about Mantra, let's look at good strategies to use it with.
First of all, it comes to no surprise that you really should only use Mantra for the shield and wave abilities, for this particular build. Mantra-shield will deal sufficient damage to enemy champs while either ganking, fleeing, bush-shielding (my term for hiding and casting on a nearby visible ally, damaging enemies by surprise), and overall ally support.
Almost never use a charge of your only two Mantra charges per cooldown on spirit bond, as this will only be as a last resort to speed an ally up or slow a pursuing enemy waaaaay down. Mantra-bond will NOT deal more damage to passed champs than it does with just a regular bond.

Karma is AP-based and very squishy, so needless to say she works well as a support with other allies.
Here are some helpful tips on how to effectively utilize Karma-based teamwork:

Offense:
-Using Flash, get closer to a defending enemy and mantra-shield immediately, then right after spirit bond them, heavenly wave immediately after, then mantra-shield damage once more while still in range. If done correctly, prepare to do massive amounts of damage and slow them down.
-When attacking a turret, be careful not to spirit bond with an enemy champ near the turret, as it will slow them down but also have the turret target you.
-Remember: Heavenly wave is VERY EFFECTIVE in dealing quick damage to multiple champions, especially ones you ran over with the spirit bond tether, and after mantra-shielding yourself to harm them even more.

Defense:
-When being ganked, immediately Flash as far away as possible. You will die from a full-on gank. If this doesn't work, mantra-shield yourself, then immediately after mantra-spirit bond the nearest enemy, and run away like the Roadrunner and watch them chase after you into a turret.
-If an ally is low on HP (and in the heat of battle) be sure to mantra-shield him to deal excessive damage to enemies around him, and keep him from dying. Also, immediately after that spirit bond him, and then mantra-wave him to not only heal him, but deal residual damage to enemies nearby. This may give him the edge he needs to flee or continue fighting.
-Always defend a turret with an ally. Karma is SUPER SQUISHY, even later in the game. Keeping this in mind, the shield is a must-have with mantra when your turret is under attack. Also, if they target you while in range of a turret, an immediate mantra-bond might possibly mean the difference between a flee and a kill.

Now, if you desire, here is a quick tip on WHICH champions Karma works best and worst with:
BEST:
Champions that aren't very fast but have high defense such as Malphite and Nasus could use your Spirit Bond effectively, and while they become targeted by an opposing champ, mantra-shielding them will not only make them much harder to kill, but also open a healing wound on the enemy, ready to be spread.

WORST:
When choosing which champions to support, it is important to take into effect how fast a champ is and what they have ability-wise. For this reason, stay away from champions that teleport, go stealth, or that have/have the capability to have high movement speed. Reasons behind these are simple: Spirit bond. Spirit bond has a very specific range that, once the tether passes that range, causes it to break. Also, when a champion tethered with Spirit bond goes stealth, the bond is broken. Even while you can see them as they are stealthed (Teemo and Evelynn as allies, for examples) you can't bond them until they are visible again. With these being the cases, champions like Shen and Master Yi with their high speeds, Tristana for her rocket ability, and Teemo and Evelynn for their stealth abilities are not ideal to support, as you will more likely than not playing on the offensive, since you can't bond your own ally, or for very long.

Karma, in this particular build, is most definitely an AP Beast. Using the above techniques, hints, and tips, you will be sure to stun any enemy with the proper practice and build. However, don't think that using this build will effectively give you an edge on the opponent, as most of this build relies on Karma as a support champion, in which you need AT LEAST 1 other ally with you to properly utilize her full ability.

PROS:
-Excellent AP build.
-Her passive, going along with the build, aids in her being easier to fight with.
-Later in the game, Karma will be able to support/gank/bash away a fair amount of champions.
-Damage being done to enemies becomes primary source of effectiveness as the game progresses.

CONS:
-SUPER SQUISHY (I literally cannot emphasize this enough)
-Above con allows for Karma to constantly be tied to an ally.
-Above con as well will not always protect you, as Karma is known to be squishy enough to ALMOST ALWAYS become the enemy's first target in a fight.
-Item build for this particular "recipe" is very expensive. Many may not even get more than 4 of the big items, even after an hour of play.
-Farming and jungling will prove to be really important, as without these you will usually stay a very low level compared to your allies (I once finished a game at level 11, while the others were 17 and 18).
-Formation of staying with an ally allows for slower leveling, as you do less damage than you might think early-game.