While playing Kai'Sa I'm sure you've noticed she has a bit of a short range, and her gap closing ability pre-six is a movement speed steroid that's only really good for dodging, escaping, or very risky gap closing since you can't auto during it. So to make up for all of this you go Black Cleaver. Movement speed from phage, some Health, some Ad, and most importantly Armor Reduction. Her Q is her bread and butter for this build, where-as her E is for the mid and late game transition. Since her Q range seems to be about 550, and does a lot against isolated targets, which seems to be the idea for most champions from the Void. Your Q on a isolated target will stack all six of black cleaver procs for a 24% armor reduction, and give you a massive burst of movement speed. This is also why you go Electrocute so you can burst them even more and get out of there.

Kai'Sa has a real problem of being blown up on a dime. That's why her core is Black Cleaver, Phantom Dancer, and Death's Dance. This will make it so you can duel almost anyone. Add a Wit's End and Mercury's Treads and you should be good to fight one on one or in team fights.

-Black Cleaver and Death's Dance is for your upgraded Q
-Phantom Dancer, Greaves, and Wit's End is for you upgraded E
-Wit's End is for your passive (since it does magic damage and is current health execute)
-Everything should focus on you surviving, or are an ADC afterall

Kai'Sa is a very strange ADC. She seems to focus around trying isolate targets around the map or in team-fights. This build ensures it doesn't matter if the isolated target is another ADC, someone who Bursts, or even a tank. You will kill who is in front of you, so long as it's one target at a time.