Ok Let's get it started...

1. Summoner Skills...

Why ignite and exhaust?

From beginning you will fight with only 1 enemy. Your rotation should look like that
((Killer Instincts))>((Shunpo))>((Exhaust))>((ignite))>((Death Lotus))>((Shunpo))>Bouncing Blade for killing.

Ignite gives you additional dmg, which can make you win.
Exhaust is spell to slow enemy and actually blind him if he is DPS.

Alternative spells:

((Flash)) - for fast escape, when you have no friend nearly or if your shunpo is on cooldown.
((Teleport)) - Fast return on the lane.
((Cleanse)) - Quick recover, when you're stunned. Remember! You are squishy so it can be good choice.
((Ghost)) - It is useful, when your enemy is out of range for your shunpo.

Spells not encouraged:
((Health)) - This spell just sucks. It won't save you anyway. If you have to die, you will die.
((Clarity)) - Kat has no mana.
((Revive)) - It sucks, cause cooldown = 9 minutes -.-
((Smite)) - Katarina can't go jungle at low lvl.

2. About Katarina

Katarina is hero with destination for TEAM PLAY. Many times if your team lame, you'll lame too, however sometimes you can own everybody, when your team suck.
She is powerfull, when you build her on Ability Power. Her shunpo makes IMPOSSIBLE Damage. If you build her on Attack Damage, she won't be as good as she is on AP.

3. Items
In the beginning you buy ((Hextech Revolver)), because you have 15% spell vamp, so you can stay longer on the lane.
After you buy ((boots of speed)), because you wanna save your money for ((MEJAI's SOULSTEALER)).
Next item is ((Sorcerer Shoes)), it is good for magic penetration... useful for tanks.
4th item is ((Rylai's Scepter))... without rylai playing with katarina doesn't make sense. It slows enemies! So if you use your ult, they will be slowed and you can kill everyone.
After rylai you have to get ((rabadon's deathcap)) (zhonya ring before) for multipling your AP. When you got ZHONYA, just finish ((hextech gunblade)). Additional magic damage is nice thing ^^. Finally you have to buy ((BANSHEE VEIL)) for immunity for stuns and other. Why banshe veil is "Must Have Item"? It is MHI, because 1st skills used on katarina are stuns, fears, taunts, snares and immobilizes. Probably you can have some extra money so buy elixirs for AP and AD.

4. How to play!

EARLY GAME

You have to go mid!
If you don't, you will be weaker and harder to feed.
She has good harrass with knife. That is why we have choosen Knife as 1st skill.
Sometimes, when your enemy sux, you can get kill or more under 6 lvl.

Mid Game
After reaching 6lvl just try stack your mejai.
If it reaches 20 stacks, you'll be easy killing machine ^^
Go to the jungle for buffs, last hit minions to get money.
Help your teammates in team fights. Remember! You have to enter, when all enemies are targeting your tank or somebody.

Late Game
Now you are killing machine, who scares all of enemy.
Your ult should make unbelieveable damage, so every enemy will run away from you ^^
Just use your shunpo, ult and get Triple, Quatra or Penta Kill!

This is my stats after game ^^