I'm picking up kaisa and I've seen a lot of conflicting stuff on how to run kaisa.
I take Fleet Footwork, usually. The added sustain and movement speed lets me trade more effectively and stay in lane longer. I feel like Electocute is kinda awkward on Kai'Sa. Hail of Blades is fun. It covers one of her weaknesses (weak short trades) and lets her get a bunch of passive marks on a target quickly. PtA is a fine choice.
Inspiration secondary. Boots and biscuits. I'd rather spend my gold on things that evolve my abilities.
For a long time, the recommended page was bugged and recommended Stormrazor first even after they reworked it to be a last-buy item in an energized build. I wouldn't recommend it.
I start AD, then build AP. Doran's Blade to start. Rush Manamune (Pickaxe first), then grab another Pickaxe for Guinsoo's before finishing Berserker's Greaves. Then I build a Nashor's Tooth to evolve both my E and W, followed by a defensive item to counter the most problematic enemy. If the game doesn't end by then, I'll usually buy a Gunblade or a Deathcap.
Against a super-fed mage, I'll swap out the Berserker's Greaves for Merc Treds and build a Wit's End for the added MR, sustain, and to keep my E evolved.
EDIT: I think one of the most important things to do on Kai'Sa (well, any ADC, but Kai'Sa especially) is to stutter step. Move between every auto, even if it's just side to side. It helps you kite, it helps you chase, it makes you harder to hit and more of a threat. I can't tell you how many kills I've gotten just backing up and stutterstepping when somebody is trying to chase me down thinking I can't kill them.
My build is in your other post. For runes I go with Hail of blades to stack the passive fast (especially usefull with Rageblade that needs to stack up first) a,d I take defenetly ravenous hunter for more lifesteal on Q (I often get people asking me how the fuck i can heal from 10% to halflife in a fight). I personally go for the precision tree second. More dmg when i am low on health (normal state in a TF) and tenacity. I hate beeing cced too long^^
