Kayn is a jungler who can start raptors without help from his teammates to get them advantage early. My favourite path is

Raptors->Red->Krugs then look for a gank if its possible if it isnt then recall and buy Hunter's Machette and full clear your jungle and look for a gank.
Another path is Red-> Raptors->Krugs gank if its possible then Scuttle and full clear the blue side jungle and then repeat it again.
There is a lot of paths you can do the way you like and the way you are going to play you can do aggresive path which is lvl 3 path. You start with Blue->Raptors->Red and then gank.You can go another way to clear your jungle depending on enemy jungler start. You can counter-jungling him or invade him but most important thing is to have fun while doing that!

A lot of players dont know which form they should get and how to get it.Kayn's Passive The Darkin Scythe allows you to get to forms (The Darkin Rhaast or Shadow Assasin Kayn). When you choose Rhaast you get more healing when use your abillities and when you choose Shadow Assasin you get more CD reduction and bonus damage.You get forms buy damaging enemy champions,fighting melee champions will give you Rhaast and fighting ranged champions will give you Shadow Assasin.You should know which to choose because if you are fighting against melee champions and there arent any ranged you will wait 4 minutes for Shadow Assasin and this will get you behind and losing the game.You should use Rhaast VS Tanks,Fighters and etc., and Shadow Assasin against squishy champions.

Kayn is very different champion and he is very fun to play! He has mobility and unique way to gank. I never get bored to play Kayn because he basically has 2 playstyles to learn how to play.
Im not a pro player and not a very high elo player so I wanted to share my knowledge about my favourite champion and hope it will help you in someway!Hope you have fun and improvement in League!