As Kayn you want to be farming as much as possible while also having small skrimishes to get your orbs for your form. In the early game you'll try to avoid the enemy jungler and focus on taking your camps. If the enemy is bad at dueling you should try and 1v1 him. If the enemy is good at dueling early such as xin zhao or kha'zix you should look to counter jungle and steal their camps.
When you reach level 6 you should look for a lot of ganks. You should not force ganks as Kayn since you will be a powerhouse in the mid-game. But that doesn't mean you should be sitting in a tent inside your jungle. A good balance between ganking and farming is important and you will learn it from playing this champion.

If you are new to this champ it can be quite difficult knowing when to pick which form. Some tips you can follow are the following:

1. Pick shadow assassin if they have 3 or more ranged characters. There are some exceptions with champions like urgot where you would rather play as Rhaast. Same thing goes for Rhaast but with melees instead.

2. Do they have more squishies or tanks? More squishies = Shadow Assassin, Rhaast/darkin = more tanks.

3. End the game quick? Pick Shadow Assassin. He is very strong in the mid game and has a lot of carry potential. The games lasts longer in low elo which is why Rhaast might be better at times (people doesn't know how to end). But I would say that the two options above are better.

4. Do the enemy have a treath that only you can deal with, pick the form that suits the best in that situation. Most of the times you should know which form to pick while in champ select.

5. In champ select, optimize for the form that suits best with your teamcomp. If your team has no frontline, go Rhaast. If you need an assassin pick Shadow Assassin.