Kai'Sa is best used - at least in this build - against very tanky teams, as a lot of her early money is dedicated to Guinsoo's in order to boost her plasma.
This makes her truly shine against tanky teams, but against squishier opponents you might lose the damage race.

You can also pick Kai'Sa early as first pick, since she doesn't have a fixed position yet - chances are good the enemies suspect to see her on botlane and arrange for that.

So far I ran it with Press The Attack, but I came to the conclusion, that Kai'Sa would likely profit more from Lethal Tempo.

Triumph was picked more for the 20 additional gold that the heal, as her gear is quite expensive, though the other options do not seem to bring anything along that would make them more interesting.

More AS means more plasma popped. Not much to say here.

Coup De Grace further augments the power of plasma against low health targets.
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Resolve - picked out of necessity to make you a less soft target - should give you that little extra edge with bone plating for ganks and iron skin for the jungle. Not much more to say here.

Second Skin (passive)
This is your main damage tool, especially against tankier foes.

Icathian Rain (Q)
Useful for waveclear, additional damage during fights and for chicken dinner at the beginning.

Void Seeker (W)
A versatile ability, which serves as scouting tool, poke and helps to take down fleeing enemies all at the same time, also it's upgrade is possibly the most important one, as it refunds 50% of the CD on a champion hit.

Supercharge (E)
Without the upgrade basically your run-of-the-mill AS booster, with some MS boost thrown in for good measure.
With the upgrade however it is one of Kai'Sa's most important positioning and escape tools, allowing her to quickly change her position more or less safely.

Killer Instinct (R)
Good in combination with her W to stick to an enemy, but an ability that should be used with caution, as it can put you in a position that is, let's say, less than optimal
Decent engage tool during laning, often useless during teamfights.

Yes. The idea here is, if you have a midlaner who starts with a decent AOE you can go for the chickens, in this case you start with your Q and the talisman. The rockets allow you to keep up the lifedrain of the talisman on the chickens, making you lose life significantly slower and the chicken super fast to clear. If it is possible, I would prefer this over method #2.
Next camps are wolves, blue and then red.

Starting with the blue - which, surprise, you do if your midlaner doesn't start with an AOE or the damage/costs do not allow it to help you or make it inefficient.
In this case top/bot helps you with the blue. You likely need more help here, try to save the smite for gromp, to quickly follow up.
In this case you rather start with the W.
Next camps are gromp, wolves and then red.

After finishing red, you should start ganking if the situation allows it. You are pretty squishy and your damage is not yet what it will be in 5 - 10 minutes, so be careful and rely on your allies as far as possible.

Use your W in order to both open as well as to shoot down fleeing enemies - the upgrade will significantly improve this ability, hence why I focused the gear choices to unlock the upgrade early.

If you managed to make a kill consider pushing the turret (if your laner doesn't oppose the idea), as you really need that sweet cash to unlock especially the W upgrade ASAP.

During teamfights I recommend you - especially if you didn't pick up a tanky option for the final item slot - to play more or less like an ADC and stay behind your tanks, try to keep your Guinsoo's stacks up and make sure you are in a position that is safe.

Your ultimate should not be used most of the time past the laning phase, as it places you usually somewhere where you don't want to be, though it is an option if a lone enemy runs away or you need the shield to survive that ignite/zed ult/kayn ult.

So, this is my first guide (ever) and I only played Kai'Sa once during an unranked SD, though I'm mostly confident that the build can hold up, though I am aware it has flaws.

I'd love to hear your improvement suggestions and I hope it provided the guidance you searched when coming here,

CPTpurrfect