This Kai'sa build can be taken any lane. Take smite of course for jungle, but do not finish your jungle item. I personally use it mid but my friends have had success in jungle and bot.

Electrocute-

Amazing for burst, lets you delete the enemy.

Sudden Impact-

Your ultimate procs sudden impact allowing you to W then R to your enemy and one shot them.

Ravenous Hunter-

This combined with Hextech Gunblade lets you heal for a large amount and keep absolute focus active.

Absolute Focus-

Gives you between 5-40 AP based on level, greatly adds to damage.

Gathering Storm-

Allows you to turn into a late game monster.

Nashor's Tooth-

The early attack speed and on hit damage from Nashor's helps with early harass. In general all of the stats it gives you synergize with your kit.

Berserker's Greaves-

Very important you get these, greaves with nashors allows you to evolve your e.

Hextech Gunblade-

This along with ravenous hunter makes your q's heal you an insane amount of damage. This item allows you to evolve your w.

Duskblade of Draktharr-

Allows you to evolve your q, its active procs off of your e, and the lethality it gives makes your q's hurt more.

Luden's Echo-

Caps your CDR and makes your w's do lots of damage.

Rabadon's Deathcap-

Practically doubles your q damage and gives tons of AP.