In the early game, try your best to farm as much as possible until you have your ultimate. Once you have it, it's much easier to kill steal- I mean secure. If your lanes are really falling behind, you still have enough threat to kill a squishy pre-6. And don't dismiss counter-jungling. Your clear speed is very high, and makes it fairly easy. Going for early dragons is a viable option, but only if you have at least 2 allies with you, optimistically one being tankier.

Around levels 8-10 is where you should get your form, 11 if you are behind/power farming. This is your main power spike, and make sure to choose your form wisely. Rhaast is best against tanks, bruisers, and bulky supports. Shadow Assassin is best versus mages, marksmen, and squishy supports. Ignore enemy assassins when choosing your form, as both have their strengths and weaknesses against them. You want to go for the form that deals with the most enemies, and if it's split 50/50, choose what your team needs.

Rhaast is a high damage bruiser mixed with a tank, so shredding theirs in the frontline is your best option. When you are low, use your ultimate on the enemy with the highest maximum health for the best healing results. If an assassin comes for your backline, feel free to use your W to CC them, and try to draw their attention. You can easily 1v1 most assassins apart from Fiora, or maybe another Rhaast if you're in blind pick. Either way, try to make the enemy tanks pull their hair out in anger, because you're allowed to burst them down as a fellow tank.

Your job is to delete the enemy mage, the support and the marksman, preferably in that order. You can either use your ult as a neat dodge, or a high damage AD burst in the late game. If any assassins are on your backline, and they are a more important target than the squishies of your team, you can most likely burst them down, unless they're a Rengar or a full AP Malphite. Rengar can heal a lot of your damage if he lives, and a full AP Malphite still has massive amounts of CC, big amounts of armor, and can probably kill you.

Q-Your Q is more than just damage: it's mobility, as well as 2 procs for electrocute if you go that route. It's also a tank's worst nightmare in Rhaast form.
W-Your W is a basic poking tool, at least as base Kayn. As Shadow Assassin it helps you kite, and as Rhaast it's a good knock-up.
E-Your E allows crazy gank paths, and has a really good heal. As Shadow Assassin, it also gives slow immunity, so it's good vs anyone with tons of slows.
Ult-Your ultimate is a massive amount of damage if you build for it, and can dodge most important CC abilities, or damage. As Rhaast, ult the tank as you watch their hope drain out as you come back out with much more health then you did before, and have a giant chunk of their health bar missing. As Shadow Assassin, your ult is best used to run down a fleeing enemy, or your passive reset.

Base-You are a gank-heavy late game jungler. Those 2 words sound weird together, but ganks mean power to you. Gank as much as possible when optimal, and you should do well. Base Kayn is classed as a Skirmisher, and is designed as such. Kayn in the early game does best at dueling other late game champions in a 1v1.
Rhaast-In this form, you are a lean, mean, tank-shredding machine, and are amazing at leaping into fights with no concern for your safety, not unlike Kled, because you have the raw healing to do so. Rhaast is considered a Diver, meaning he is a mobility-based fighter. He fulfills the role quite well.
Shadow Assassin-This form is based around pure, raw burst, with a nice bit of sustained damage thrown in as part of the AD package. You are similar to Akali in the sense that you dash both into, throughout, and out of a fight. As suggested by the name, you are classified as an assassin, and should be treated like one. You shouldn't completely disregard safety, just take it into less of an account, because unlike other assassins, you have a good heal if things go sour.

I was heavily inspired by Bricky when making this guide, and I hope I got you some nice LP. See you, and try not to feed in your promos.