This is my build for the Katarina. It's a nice build of damage and spell vamp, with the progressive AP glyphs that will keep you gaining AP as long as you're keeping up with levels. Tell me what you think in the comments below and make sure you vote!

You can switch out the progressive ap glyphs for cooldown reduction if you want. Also you can switch out the AP quintessences for health if you need the survivability rather than the damage. Other then that the magic penetration

I left out one item for you to choose your on your own. The rest of the items I feel are pretty much what's needed for a lot of damage, health, and spell vamp. If you end up doing enough damage by the time you get all the items, you might want to get a defensive item such as Guardian Angel, Force of Nature, or Warmog's Armor. If there is a tank you can't quite nuke down, try getting Void Staff for the AP and magic penetration. I normally don't get Mejai's Soulstealer because I try to get the spell vamp items and Rylai's Crystal Scepter pretty fast for the survivability, then it's usually either over or too late in the game to consider Mejai's, but it's up to you. The Will of the Ancients will give you spell vamp, and other people on your team spell vamp and ap, helping them out a bunch.