This build for jungle kai'sa is a little different, I don't think anyone has Jungle Kai'sa figured out yet, jungle Kai'sa is a beast. in Silver 5 at least

One of Kai'sa's worst attributes is her speed, if you get caught out you are dead, you have no real escape and no really accessible shield unless you turn around and try to fight. most people say take fleet footwork for the speed because phase rush doesn't really help much. I take predator, the boost of speed is helpful for both rushdown and for escape. Once your E evolves the invisibility can buy you just enough time to pop predator off to escape or to turn and fight. For secondary tree I take sorcery for the move speed bonuses.

The most important thing for Kai'sa is her choice of items, her passive gives her a boost to each ability once you hit a certain skill cap.

The jungling items offer kai'sa a great option there are 2 good option that I find on Kai'sa with this build, Warrior rune, and Bloodrazor. Warrior will evolve your Q with gunblade, Bloodrazor will evolve your E with Berzerkers and Rageblade.

Gunblade and Rageblade - yeah its a weird combo but it makes sense, with both of these you have built enough AP to evolve your W, gotten a good start to your attack speed and built up 90 AD, thats 350 gold for a long sword away from evolving your Q. Not to mention with both of these items you force the enemy team to play around both your AP build and your AD build.

Lethality is really strong for quick kills, your E evolved lasts just long enough for a duskblade proc (until late game), or take ghostblade for the boost in speed.

Crit shouldn't really be a focus for you as Kai'sa it's nice, but your passive helps you pop squishy targets after a few hits because your plasma does damage based on missing health.

BORK is insane on kai'sa you do damage based on how much health a target has, and because of your passive also how much they are missing, double whammy.

OH GOD HOW DO I GANK

it's easy you just have to put in the WERQ
W->E->R->Q

Kai'sa's Q is your main damage spell, but you need to be hugging your enemy to hit them, (though that hug may not be so nice, Kai'sa BAD TOUCH) taking this at level 1 you can kill red or blue buff with half of your health left without needing a leash or having to smite.

W a SUPER long range projectile that moves with meh speed. This ability does damage too so that's cool I guess... we take 1 point in this early and don't really touch it until the end. Most of the damage comes from scaling anyways which (for now) is flat scaling 150% AD and 65% ap. which is a nice bonus but ill talk more about why this ability is useful later.

E is kind of a weird ability, at first, it gives you a burst of movespeed kinda nice, but then it gives you bonus attack speed, REALLY nice. Evolved this gives you invisibility too which I mean I'm not going to complain but like meh. plus the more attack speed you have the less time you actually are invisible or being a blue hedgehog... bug... girl.

R Dashes you to an enemy champ marked with plasma, plasma is built up when you hit a champion and at 5 hits pops for bonus damage. this also offers you a shield that scales with AD and AP.

Alright you know how you didn't read my paragraph on your W and talk about how it will be useful and I'll talk about it later, here is where I talk about it. W applies plasma, see where I am going... no... fine. your W has a stupid long range, this lets you hit enemies that are running away, OR gank from like half a ****ing lane away. someone thinks they are going to get a kill for free NOPE. launch your W towards the enemy, pop E in case you miss or if you hit you have your attack speed buff ready. once it hits use your R to surprise the enemy drop a chilling smite and mash the Q button to nuke an enemy.

Blue - gromp - wolves -red - raptor - gank

Kai'sa doesn't need a leash for level 1 buff

KAI'SA DOESN'T NEED A LEASH FOR LEVEL 1 BUFF

really your Q does enough damage, you don't even need to smite it's that good.