Not Updated For Current Season
This is my kat build.
1.just like is most games you need to rush your occult and soulstealer.
2.right after you get sow/ms u might want to get sorcerers shoes. Before you say "WTF FAIL WHY WOULD U WANT SORCERERS..." it gives you penetration and ALL of your moves are magic.
3.Bloodthirsters go awesome with kat so get one.
4.Rylais slows enemys gives you health and ap...so why not? It helps with ur ult once you shunpo on that should slow them so you can get more of your ult on them...and your ult should slow them as well.Also I dont think the will be going anywhere if you put exhaust on them.
5.The last items is up to you. But i usually go for a black cleaver.
This is my Katarina build.