Kai'Sa is the newest champion we know of in the League of Legends universe. We don't know exactly what she's good at yet (even if she seems to be assigned as an ADC Bot Lane). So why not try her in a different lane? Sure, some characters aren't meant to main a lane they aren't assigned, but since Kai'Sa's abilities are affected by her build, that means you could technically take her anywhere until proven otherwise. Mid lane isn't the easiest: you don't have a designated support unless the jungler wants to help you, and you can be ganked from the river on both the left and right sides. But I think it wouldn't hurt to try it out.

Passive: Kai'Sa'ssecond skin allows her to be a "living weapon", which allows her to 'upgrade' her Q, W, and E. This can be done by building either Ability Power, Attack Damage, or Attack Speed.

Q: Kai'Sa's Q can be upgraded when you have enough additional attack damage (or AD). By choosing cores like Bloodthirster and Hextech Gunblade, I achieved +120 Attack Damage, which is more than enough to upgrade from 6 missiles fired to 12 missiles fired.
W: Kai'Sa's W can be upgraded when you have enough additional ability power(or AP). By choosing cores like Hextech Gunblade and Guinsoo's Rageblade, I make my way up to over 100 additional AP.
E: Kai'Sa doesn't necessarily need more attack speed for her E, but I wanted to try out building attack speed anyway to see if it affects her gameplay positively, especially in mid. Adding Precision Runes like Press the Attack might elevate this even further, so in theory, I might be able to reach high attack speeds or at least attack speeds over 1.7 or so.

To upgrade your q, which will prove helpful in the beginning of the game for farming purposes, Icathian Rain needs 100 AD besides your base. This means you want to start building AD. For this build in particular, I'm choosing to build AD, then AP, along with some experimental cores that can be swapped depending on what you want to upgrade in particular.
The first core you want to build is Bloodthirster. This spikes your AD by 80 while giving you lifesteal so you can survive longer in lane without having to constantly go back. After Bloodthirster, you need 20 more AD, and you might as well start with AP, which upgrades your W. Hextech Gunblade both completes your AD requirement as well as spikes your AP. Now you have +120 AD and +80 AP. Needing 20 more AP, go for Rageblade. It'll both complete your AP requirement and spike your AD yet again. Also, you have attack speed slowly building up with Rageblade, which is helpful when you want to use your E. I added crit and more attack speed with the stattik shiv. Again, this is an experimental build, as Kai'Sa hasn't come out yet. I chose the Cores i did to balance out AP and AD, as well as add some extra attack speed and crit for a bit of boosted fun. Boots you do need, but after getting boots themselves, your pick in upgrade is based off preference. You need attack speed? Do it. You need extra Movement speed? Do it. Do whatever you want with your boots (and if you want to substitute one aspect of this build for another, like crit for armor, do it. It's your build. I'm just giving you a suggestion to elevate what seems to be the best aspects of Kai'Sa herself. Have fun!

Kai'Sa's ultimate (R), allows her to dash to an enemy champion, even if that champion is behind or obscured by other champions, as long as the champion selected has plasma.

Plasma is caused by Kai'Sa's basic attacks, Void Seeker (W), and nearby allies immobilizing effects. Plasma lasts for 4 seconds and can stack 4 times.

Flash: I chose flash because flash is needed in almost any game unless you're Zoe and can take up flash with your W. In any case, Flash is basic. Flash is necessary, and Flash can and will help you when you need a quick fix to get away.

Teleport: Teleport can be replaced with Ignite, and vice versa. Depending on your play style, you might want Ignite to intimidate your enemies or to deal that last bit of damage when you run out of your ability resource. Teleport also has its pros. For example, if you're losing a turret fast and don't think you can make it by foot or flight, teleporting will most likely take a lot less time, despite its cooldown.

Choosing press the attack may seem pointless right now as we don't know Kai'Sa's full extended powers as of yet. You are more than welcome to change things up. I don't necessarily see the point in going for a primary sorcery rune or domination rune as of yet, so I stuck with precision.
I chose press the attack because I wanted something that was neither AD or AP based primarily. I did, however, go for a secondary domination rune, for a little extra health when I damage enemy champions and the like. You could also primarily go for Resolve or Inspiration runes, though I'm not sure how that would turn out. I could see a primary Resolve rune working, but I can't really see the pros right now for mid lane, as I'm not trying to build tanky nor am I trying to push for more maximum health then my enemy champion.

First of all, have fun. I don't think anyone is going to get the hang of Kai'Sa in their first game or training session. It will take a lot of experimentation (like how I am going to try the mid lane position with her), as well as patience. My build may not work first try, that's the truth of it. You may have to substitute crit for armor, or attack speed for attack damage, we don't know yet. The only way to find out is to try something. And if that doesn't work, play around a bit, find her niche, and work with it. Good Luck!