Intro:

U have to know how you play Kata, i play kata always as a mid champ.
If u dont like her mid dont worry this build works everywhere.

Kata has no mana or energy etc.
You can stay long on your lane.

Kata is a lil bit squishy but thats why i get the Rylais.

You should take the Greater mark of Insight because of magic penetrate. It is nice
when you fight vs a tank.
And the Quintensseces, Seals and Glyphs are AP to do more damage.

Idk why i have this masteries, i had them from Nidalee Build and was to lazy to change but it works also good with Kata.

I take the Sorcerers Shoes because of the Magic Penetrate.
Hextech Gunblade is my first choice because it has spell vamp so u can stay longer on ur lane, it also gives AD and life steal.
Rabadons because it gives much AP.
Rylais to give Kata more hp and slow.
Void staff because of the magic penetrate.
and another Rabadons because the AP, what you also can do is take a Zhonyas Hourglass.

I would take ignite to finish them off.
I would take exhaust if u jump in to exhaust them that they cant run away for your ulti.