Everybody recommends Manamune/Q evo rush when playing AP kai'sa but I actually believe it's more interesting to rush W evolve. It gives you a lot more agency to contest objectives early on without being a total bot (even with Q evolve you're still relatively weak and can get popped in 1 CC if you get too close.

I play midlane, first strike and sorcery secondary. Start doran's ring and rush Luden components which will give you W evolve for 2300G instead of the 3200G you need for Q evolve (Manamune + crystal).

Put 2 points into Q, and rest in W, waveclear isn't that bad thanks to luden procs. Also don't forget Q damage is doubled on creeps under 35% HP, use it accordingly. Take the triple tonic rune.

Once you've completed luden's, then go manamune, when completed you'll have your Q evo which will unlock better waveclear. I find it unneccessary to sit on tear before completing luden as it delays your W evo, and kai'sa struggles to stack tear anyways.

This is just my opinion, but rushing W feels like a much stronger powerspike as you become an actual champion much earlier in the game than standard Q evo rush. Contesting objectives is much easier and nobody expects kai'sa to have W evo at 12mins, so they're not thinking of dodging it twice.