I was thinking about take lich early, but if u r doing really well u always will be focused on team fights, zhonya is really great for escapes, and GA can help u on team fights too...
But I was wondering if GA is necessary after Zhonyas

About dorans, he can help u a lot early game with health regen, and hp

I'll try your build

I think that is not so bad, but... doran shield at start is useless. Lich bane is a good buy as the 4th ITEM not as last. GA is great for kat, but i think when you got one great tank, you dont need it. Sometimes you need to buy Mercury treads and you are in *** with your magic pen. So... better is Void staff than GA.

My build: http://www.mobafire.com/league-of-legends/build/katarina-im-overpower-ap-27906