Karma is an AP carry with a couple support skills, same as Lux or Orianna. Don't let anyone tell you otherwise.

Magic pen red and quints for extra damage, mana regen/lvl for lane stayability, flat CDR for better teamfighting. Quints can be swapped for MS or flat health if you don't have any magic pen.

Standard AP Carry 9/0/21.

Start Doran's Ring. Magic pen boots (if you can't afford them get normal boots and another Doran's). Rod of Ages. Deathcap. This is your core build. After this you should counterbuild the enemy team. If they are magic heavy, build an Abyssal. If they aren't focusing you, get a Deathfire Grasp or a Lich Bane. If your team is magic heavy, get a Will of the Ancients. If their team is AD heavy, get an Hourglass. CDR can also be nice. Get a Kage's lucky pick early if you can, GP5 items OP.

Some people like Rylai's on Karma: I don't, her Spirit Bond is enough of a slow.

Skills:

Karma sends forth hidden blades from her fans, dealing 70/110/150/190/230 (+0.6) magic damage to units in a cone in front of her.

Mantra Bonus: In addition to dealing damage to enemies, Heavenly Wave will also heal allies in the cone for 35/55/75/95/115 plus 5% (+%) of their missing Health.
Cost: 70/75/80/85/90 Mana
Range: 400

Don't waste a mantra charge on this skill unless your allies/you are at very low health. This will not heal for very much unless you are at low health. If lots of your team are at low health, it's worth telling them in chat to stand next to each other. Most people won't think to do it on their own.

Spirit Bond - Karma creates a beam between an ally or enemy. Allied anchors move faster and enemy anchors are slowed. The beam deals magic damage to enemies and applies the same movement speed adjustment anchors receive to any champion it passes through.

Mantra Bonus: Karma strengthens the bond to double the effect of the movement speed modifier.
Karma creates a beam between an ally or enemy for up to 5 seconds. Allied anchors move 10/12/14/16/18% faster and enemy anchors are slowed by 10/12/14/16/18%. The beam deals 80/125/170/215/260 (+0.7) magic damage to enemies and applies the same movement speed adjustment anchors receive to any champion it passes through.

The beam breaks if the bonded target is stealthed.

Mantra Bonus: Karma strengthens the bond to double the effect of the movement speed modifier.
Cost: 65/75/85/95/105 Mana
Range: 800

Never waste a mantra point in this. Take one point in it at level 4 and use it to slow enemies to kill them, to escape, or to get to your lane faster.

Protip: use this on a ward for a free WTFescape.

Soul Shield - Karma summons a protective shield that absorbs incoming damage.

Mantra Bonus: In addition to casting the shield, energy radiates out from the shield, dealing damage to enemy units around Karma's target.
Karma summons a protective shield that absorbs 80/120/160/200/240 (+0.8) damage for 5 seconds.

Mantra Bonus: In addition to casting the shield, energy radiates out from the shield, dealing 80/120/160/200/240 (+0.8) magic damage to enemy units around her target.
Cost: 70/80/90/100/110 Mana
Range: 650

Use this whenever anyone tries to harass you in lane, mantra it if they are in range. If you can afford to, use it to deny last hits. It takes some reflexes to use this to stop an Annie fireball or similar skills but if you can you will be super annoying to lane against.

Protip: Use this on a ward for a free WTFkill.

Mantra - Karma empowers her next ability to do an additional effect. Mantra is available at level 1 and does not require a skill point.
Karma empowers her next ability with an additional effect. Karma gains a charge every seconds up to 2 charges.

The refresh rate is reduced by Cooldown Reduction.
Cost: No Cost
Range: 200

Fairly self-explanatory. Press R and then use a skill, it'll have a sweet bonus. Don't forget that CDR will make your mantra recharge faster!

Inner Flame - Karma gains increased Ability Power corresponding to her % of missing Health.

This passive is very strong at low levels, it's worth staying at low health just to get the extra AP.

Skill sequence is mostly up to you. I like to get a point in Spirit Bond at level 4 so I can slow people to get my other skills off on them, but Heavenly Wave and Soul Shield are both fine to level, max whichever one you want first, I like to level them up about equally.

I like Ghost Flash. Flash is fairly necessary on Karma for both securing and escaping from kills, but Ghost can be substituted with CV, Teleport, or Ignite.

Take mid if you can. Solo top if it's against a squishy/unsustainable champion. Don't go bot with AD carry, you're not a support, you need farm.

Laning:

Harass your opponent with Heavenly Wave and Mantra'd Soul Shield. You want to be using your Mantra charges almost exclusively on your shield, to do maximum damage output you should stay at around half health, only use a Mantra'd heal if you're at low health and think the enemy laner can burst through all of your health and shield. By staying at low health you are proccing your passive and baiting the enemy into attacking you, at which point you can burst them with passive procced Wave + Mantra'd Shield while not taking damage yourself with shield. If you are beating the enemy in lane you can zone them easily, if they try to last hit you can use Spirit bond to slow, then your other two skills to burst them down. Karma is an aggressive laner, you will usually be very pushed so make sure you buy wards to prevent deaths from enemy ganks.

Teamfights: You want to be on the edge of teamfights, just in range but not in danger. At early levels you can put a non mantra'd shield on your tank before he initiates, at low levels he will be in danger because he doesn't have many tanky items yet. While in teamfights, try to put mantra'd shields on your tanky dps/melee dps, spam mantra'd waves into teamfights to heal allies and damage enemies, and either use your bond to save teammates or slow escaping enemies. Use your flash to pick up kills or escape if you need to. Don't be afraid of "killstealing". You're a carry, not a support. Don't let anyone try to convince you otherwise.

Ganks: I prefer to stay in my lane and get valuable farm, but if you are 100% sure you can get a kill, or if your allies are losing a lane, you can go gank. start off with a spirit bond to slow, then heavenly wave, then mantra'd shield to nuke + tank the tower.

Jungle: If your lane is too pushed to farm safely, go jungle. Take blue whenever you can, and feel free to kill wolves, golems, and wraiths, you will be able to get lots of farm without taking damage. The little camps respawn fairly fast, your jungler shouldn't be too pissy as long as you are already fairly far into the game.

Pros:

Great burst
Large shield
Good at chasing down a target
Good at escaping if near allies or minions
Passive gives a lot of extra damage, especially at early levels
Scales well through all stages of the game
High skillcap (I count this as a pro, easy champs are boring to play)
Many people underestimate this champion

Cons:
Weak heal
Heal can be very hard to land on multiple allies
Only 3 abilities
No ult
Spirit bond is hard to do damage with
People will queue dodge or rage at you when you try to play her