18/09/2011 Added some things, deleted the DPS part of the guide

Welcome to my first guide! You'll find a lot of typos (My english sucks) but I'll try to make it the best I can.

The AP build is similar to the recommended, but there are 2 changes, instead of Doran's shield we get the Void Staff for more Magic Penetration, and we get Morello's evil tome to reduce the cooldowns (The mana regeneration thing is useless, though), if the other team has a lot of AD champions, you can replace it with a Frozen Heart.

I created the DPS hybrid build just to steal some life from the minions, and it actually worked very well, but it has some flaws.

Pros:

Fun to play
She's good at ganking
No mana

Cons:

She's squishy
Needs to farm a lot to become useful

I reccomend Exhaust and Ghost, Shunpo replaces Flash and has a lot of range (That's why I don't pick it)

Alternatives: Flash can be very useful.
Heal helps a lot at the first 10 mins of gameplay and if you use it wisely it's useful later too.
Cleanse helps when you get ganked and someone stuns you.
Ignite for those moments when you're slowed and an enemy is getting away, you scream "Oh no! You're not getting away!" and use Ignite. If there's a Warwick on the other team
Spells you shouldn't use: Clarity, do I have to tell you why not?
Clairvoyance is useless with this champ when you communicate with your team.
Teleport: It doesn't help at all.

Against an AD team

I reccomend replacing a Rabaddon with a Thornmail or a Frozen heart (Because of the 20% cooldown and Attack Speed reduction)

Against AP team

Replace a Rabaddon with a Force of Nature.

Alternative Items

Rylai's crystal scepter can be useful if there are no champions with CC in your team.

I like to buy Morello's Evil Tome sometimes because of the Cooldown Reduction.

Best laning partners

Garen: Garen can silence any champion who can cancel your ult (If Kat gets stunned, silenced, knocked or taunted she will stop casting her ult).

Soraka: I know what you're thinking. "What, Soraka? What he f**k were you thinking?". But she's actually pretty good at helping Kat, she can heal, silence enemy champions, and reduce their Magic Resist (Remember: Kat skills are AP based)

Shen: Because of the taunt, while you're casting your ult shen can take any enemy champion to it.

Jax: His stuns are useful, if you have some coordination, you can Shunpo at an enemy when Jax stuns him, throw Bouncing Blades and if the enemy is still alive (I doubt it) you can reuse Shunpo or you can use your ult.

Jarvan IV: When Jarvan uses his ult, Shunpo in and you'll get at least one kill. He has a lot of HP and Defense, if the person playing him is good, you'll always be safe.

Worst Laning partners

Other squishies.