Items
Kraken/Shiv->Guinsoos->Nashors->Flex options depending on enemy comp
Explanation on when to go which Build
AD vs AP: This mainly depends on your teams damage share, you want to balance it out so if you have 2+ Champs that deal magic damage, do NOT go for AP builds. If you have less than 2 AP damage sources you can decide for the AP builds (Id generally go for this since Kai'Sa AP scalings are nicer than AD scalings and AP items are also more convinient to build.
Shiv vs Kraken: in all the builds I showcased you can also decide to go Shiv instead of Kraken, the benefit of this is that Shiv deals some extra magic damage and is cheaper than Kraken. The most important thing is that Shiv provides faster waveclear -> more tempo to roam on sidelanes or just to leave pressure vs teamcomps that can siege a lot while Kraken provides more DPS vs tankier teams/ teams you can autoattack a lot against
Now that we know when to go AD and when to do AP builds there is still to decide: If we go AD: Crit or OnHit AD? Very simple Crit = more DPS while OnHit AD allows for more resitances/tankyness So if you feel like you cant survive for example vs Burst/Assassins go for The OnHit AD option that allows for Terminus Jak'Sho to provide you with Health, Armor and Magicresistance, otherwise just go Crit
If we go AP: Poke or OnHit Also very simple right now Poke scales too slow for SoloQ in my opinion and you should opt for OnHit AP 90% of the time, Poke is only really good if you know for a fact that you can scale until 2-3 Items and/or if you absolutely can't autoattack vs the enemy teamcomp
Important to note here is that the Pokebuild is the strongest build with the least counterplay if you get to that point of the game as having Nidalee Spears from 3x the range and bigger hitbox is absolutely broken
Matchups / When to Pick
In general you always want a Melee support with Kai'Sa as the whole champs identity is Bursting/ All-Inning which Melee supports help a lot with
Best Supps: Naut, Rell, Leona, Rakan, Pyke, Braum, Blitz, Alistar,
Hard Matchups: Samira, Draven, Caitlyn, Ashe, Varus
Mediocre Matchups: Miss Fortune, Kalista, Kog'Maw, Nilah, Tristana, Lucian, Ezreal, Vayne, Senna, Xayah, Zeri, Aphelios
Easy Matchups: Twitch, Jhin, Smolder, Sivir, Corki