Kayn can clear the jungle really fas which he will use to get a lead and snowball. You should try to get your laners ahead and invade to delay his power spike.
His allows him to have really unique ganking paths, he can easily avoid vision and surprise your laners. Have a mix of deep and shallow wards to reduce his gank potential.
He has a really weak early game and if he gets behind he's going to be useless. Try to get him behind through ganking frequently or invading and taking his jungle.
Shadow Assassin will look to nuke your backline by flanking or going through walls. Try to peel your backline or set up vision so that you know where he is at all times.
Rhaast will look to peel his carries and tank as much damage as he can. Grievous wounds is a must have vs this form.
The best way to win a teamfight vs Kayn is to lock him down, both forms are pretty easy to deal with when they can't move.
Kayn has a pretty strong level 2 and has a lot of sustain due to his kit. He can easily cheese objectives or go for early ganks because of this.
Kayn when he gets either of his forms is really strong. His ganks and jungle clear become a lot better and he can also do objectives easier. Try to delay his transformation by only fighting him if he is low.
Kayn is really easy to kill without his , because he loses a lot of escape potential and damage.