Kayn's jungle is not the best early, but it is still rather good. Always start Reaping Slash in the jungle, whether you are going to max it first or not, because of the massive damage bonus to monsters early game. Generally, you would want to start Raptors -> Red Buff -> Wolves -> Blue Buff -> Scuttle Crab -> Gank Lane. However, starting Red/Blue Buff can be a good idea if the enemy team has an invading jungler, such as Shaco or Lee Sin. Remember to use Shadow Step as a healing ability when low. If you are against a slow jungler, you may want to invade, as Kayn's mobility makes him very slippery. Additionally, while in combat, don't try to go through any ways that are extremely thin, as you will just be sent back to where you were. Your Shadow Step gives you a near infinite amount of choices when ganking. Avoid bushes, as those are frequently warded, and instead, use Shadow Step to gank the enemy from unthinkable places.

Kayn's two forms do both of their jobs very well. Having a neutral stance in masteries and runes is an essential part of this build so that they can both be viable no matter which form you choose. If your team already has many assassins and is in need of a tank, go Darkin. If your team is tanky but has few assassins, go Shadow Assassin. Since almost every team will have both squishies and a tank, it's very hard to go wrong with either form. However, getting one form can be difficult. You should expect to get one form at anywhere from 12-16 minutes in the game. Camping a lane that is Ranged or melee in a game is a great way to get a form quickly if said range type is rare on their team. Avoid fighting Pre-form as Kayn is very weak then.

The combat style for each form is rather different, and you will want to follow accordingly:
Shadow Assassin should be killing their backline and carries, Using the speed of shadow step to quickly hunt down the enemy team. Engage with Blade's Reach, as it has no cast time, and then Then activate thunderlords with reaping slash, Finally, use your ultimate to finish off the target, and if they survive, autoattack them once more and use abilities if they are up.
Darkin likes to attack the tanky, frontline champions. Engage with Reaping Slash and use Blade's Reach to stop your target from running away. If you ever get low, cast Umbral Trespass on an enemy tank to heal for a massive amount