Katarina is an awesome champion, with her mobility and crazy damage she can be a mass destruction weapon if u know how to do it. Huge AP / AD damage and agressive playing style make her the best char for you, not you with the yellow shirt, you from behind, you're nice =)

Normal Build

Focused on damage and mobility, max Shunpo first, because it has reduced CD and increased damage each level, let Killer Instincts level 1 for the bonus effect on Bouncing Blades. Bouncing Blades is your better harass skill, jump up to 6 targets and deals crazy damage.

Alternative

If you cannot use Shunpo to hit your enemys for some reason, level Bouncing Blades first, than you can harass youur enemy from a safe distace.

You can see that I love Hextech Gunblades =P , awesome item for Kat because of the AP, AD, Life Steal and Spell Vamp, focusing on the Bouncing Blades and the Death Lotus damage, when Rabbadon's Deathcape give a bit boost to your ability power.

Priority Itens

Hextech Gunblade - Boost your Bouncing Blades and Death Lotus damage and give you survibility.

Rabbadon Deathcape - IMPORTANT FOR AN AP BUILD, boost all of your skills...

Rylai's Cristal Scepter - Give you HP, AP and 35% slow effect with Death Lotus ( Even been one multi target skill it gives 35% not 15% slow effect )

Try to use your Flash to run or get some kills, DO NOT SPEND IT FOR A SINGLE KILL, it's important, it can be replaced with Ghost Walk but I really dont like it...

Use Exhaust with your Death Lotus and the effect of Rylai's Cristal Scepter for a instant kill, obviously better use it on the target that has more HP...

You can get kill pretty easy, with your Voracity, that is one awesome skill, if you kill 2 enemys with you Death Lotus it instatly refreshs the CD.

Pros
HUGE damage
Awesome harassing
Great 1vs1 fighter

Cons
Very squishy early game
Can be dominated with CCs