Ability Power Kai'Sa is the way to go (duo-lane)
This build does so much damage in the late game, remind you all that Kai'Sa is a hyper marksman so your power comes from when you have 2-3 items which is approaching late game, but when you get to that point to murder everything in you path!
For masteries there are a few options you can go between the 3 hit proc into bonus damage for your team, the charges that heal and grant movement speed, or slow augment depending on what you want. Personally, Slow Augment is my personal favourite to use since the Biscuits help with sustain for the early game which you need to get through some how and the slow augment sets up your passive nicely for multiple hits in into the % missing health proc, keeping them in range for longer - I also personally like magical boots and 5% CDR cap - again personal choice and you can go any of the three keystones and the other masteries.
BUILD IS: Nashor's Tooth (core damage) Guinsoo's Rageblade (damage multiplier) Hextech Gunblade (mostly for sustain since only thing that heals from on-hits, but the slow is nice too)
This is your core three items to get:
You also want the attack speed boots because you can get both W+E evolved from Nashor, Rageblade, Boots.
OTHER ITEM OPTIONS: Runaan's Hurricane (area of effect for passive and reduces E's cooldown a lot faster) Rabadon's Deathcap (MOAR ABILITY POWER, MOAR DAMAGE) Banshee's Veil (defensive item) Wit's End (mix of damage and defense) Zhonya's Hourglass (not recommended since your damage is from basic attacks and not cooldowns, hence no down time for damage, but if you are against a Zed or something this is an option)
DO I NEED PENETRATION?: No, because you are dealing mixed damage so penetration is inefficient at half effectiveness, your Q still does physical damage and dealing a lot of physical from every attack as well.
Your magic damage is still higher than your physical for Kai'Sa (same if you go attack damage, but you might as well play a completely different champion then, play Kai'Sa if you want to play an ability power marksman). So if you want a penetration item then DO NOT get Last Whisper items, get Void Staff instead since your majour tank busting damage (and most of your damage) is from your passive and magic penetration buffs up your poke damage from W.
Do not worry about evolving your Q. As nice as it may be to have around, it is not that big of a deal to not have it, E is the most important evolve because of the invisibility, but you will be evolving W first every game since Nashor's Tooth and Amplifying Tome for Rageblade and Gunblade grants 100 ability power, and then you can get E evolved from attack speed boots and Recurve Bow with your Nashor's Tooth.
W is a better evolve anyways since it offers poke damage with it for landing the spell on champions, halves the cooldown when evolved for doing this, while still getting more damage out of it since it applies three passive stacks rather than two.
Remind you that Kai'Sa is NOT Vayne nor an assassin. You are a late game hyper marksman (redundant statement there), reliant on your team to do well, and super strong in teamfights and absolutely massacring the enemy tank line!
Also remember that you should almost never go in with your ultimate, use it defensively to keep yourself alive and keep dealing damage!
Hope you all enjoyed this and have a nice day everyone! [sg-kiko]