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.

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.

Under this chapter, I will be discussing buffs, nerfs, and changes and how they affect Kayn.

10.5

Made orb gain rate a little higher over time, and a small buff towards base Kayn and Shadow Assassin by increasing the W's slow by 10%. These buffs aren't going to affect his place in the meta by a lot, but hey, a Kayn buff is always welcome.

10.6

Dropped with another Kayn buff! This change increases the orb gain rate at 10-13 minutes by 15%. This buff isn't all that useful since most of the time, you will already have your form by the 10 minute mark. But in some scenarios where ganking isn't possible, late skirmishes can give your form much easier!
The new Death's Dance is here, and will probably be very useful on Rhaast!

10.9

HUGE changes came this patch. His Q's bonus damage applies on to minions, his Q cooldown got reduced, and his W's slow got increased to a 90% de(kayn)ing slow. His Q cd buff will increase his clear speed, and fighting. The slow will be good allround, and his Q bonus damage might be enough to push him into a top- or midlaner.

10.10

Ah... Rhaast's passive healing got nerfed by a tiny amount. Since you most of the time go death's dance, conqueror, and ravenous hunter, it really won't affect all that much.

10.13

Rhaast got indirectly nerfed this patch with the conqueror nerfs. The keystone is still good but it is weaker than before, it won't affect him too much.

11.17

They nerfed Shadow Assassin's passive damage by quite the margin. The nerf will not make blue Kayn unplayable, but it will hurt his damage output.

The new XP changes in the jungle gave us more options for pathing. You can get lvl 3 from different ways now.

Recommended clear:
Raptors -> Golems -> Red -> Blue -> Scuttle -> Gromp -> Wolves -> Raptors -> Golems -> Recall
This path gives you level 5 and about 1300 gold, which lets you buy Stalker's blade (blue smite), long sword, and boots.

Examples on fast lvl 3 paths:
1. Red -> Blue -> Gromp (or Blue -> Gromp -> Red)
2. Blue -> Gromp -> Wolves
3. (Red side clear) Red -> Raptors/Golems -> Golems/Raptors

Examples on other paths:

Red (leash) -> Golems -> Raptors -> Wolves -> Blue -> Scuttle crab or gromp

Raptors (no leash) -> Red -> Golems -> recall and buy machette or long sword -> Wolves -> Blue -> Gromp or Scuttle crab

Wolves -> Raptors -> Red -> Golems -> Recall -> Blue -> Gromp/Scuttle -> Gank or Invade

Pros
Good clear
Good sustain
Ultimate and his mobility helps to dodge abilities
Strong with either form
Can choose if you want an assassin or a tank/bruiser
God tier ganks with his E
Good late game
Hyper carry
Cons
Bad at contesting scuttle
Not the best in early duels
Weak early game
Almost guaranteed that the enemy will get the first drake
Rhaast is kinda bad without black cleaver
Bad if behind
Mana issues
Needs items to be strong
If you can't get form before 13 min he struggles if not ahead
Wait 4 min if you get wrong form

Ganking as Kayn is one of the things I like the most about this champion. Other junglers can gank from a limited ways, Rek'Sai can gank from a couple more, but Kayn can gank from literally anywhere thanks to his E Shadow Step. This is one of the core things you have to learn if you are planning to main Kayn.

A couple of paths you can take to gank lanes are the following: if you are planning to gank top and you are on the blue side you can use the baron pit as a start with your gank. It often guarantee a flash or a kill since the enemy often wards the brushes and not the pit. If you are on the red side you can be across the river (right where the blast cone is located) and gank from there. These things are the same on the bot side but with the dragon pit if you are on the red side and the "blast cone area" if you are on the blue side.
Remember, hitting your w is very important in the earlier ganks of the game since it slows the enemies hit. You can also very easily tower dive if you have level 6 and good teammates of course.

IMPORTANT TIP: YOU CAN SMITE WHILE YOU ARE INSIDE AN ENEMY WITH YOUR ULTIMATE!!

YOU CAN ALSO AUTO ATTACK WHEN YOU'RE W IS CASTING (SHADOW ASSASSIN).
Transforming into either Rhaast or Shadow Assassin gives you the homeguard speed buff, therefor you should transform near the nexus turrets (you will be able to travel further).
When you take your red or blue buff you can leach it out a little bit and then use your Q in to the wall where the buff is. Since hitting a bigger wall will not make your champ dash as far but will still deal the damage. This will help you clear it faster and take less damage from it.
One of the most important things I see a lot of Kayn players do is not knowing how to use their Shadow Step correctly. If you are clearing camps you can use it to gain some hp and get around the map quicker. You can use it from the raptor camp to your blue or wolves if you go through one of the mid towers. But if you are stealing the enemy's camp or taking the scuttle crab save your Shadow Step so that you can escape if they come. Kayn is weak at dueling and you should there for save your Shadow Step when you counter jungle.After you finish your red buff and golems, use your Shadow Step to get to other camps faster and heal. When you do your raptors you should be level 2 and have your Reaping Slash and Shadow Stepunlocked. Auto attack one of the smaller first and then use your Reaping Slash on every raptor so you hit with both the dash and spin. Then leach them around in circles so you don't take any damage from them. When your Q Reaping Slashhas come up again, use it on the whole camp and only the big raptor should be alive.
If you are engaging an enemy you can use your R Umbral Trespass to dodge an ability that deals a lot of damage or stun you. You can also use it to catch a faster enemy like Rammus or Hecarim with Blade's Reach or Smite and then your ultimate. This works best if you are shadow assassin since it gives your ultimate more range.
Your utlimate is probably your most important abiliy as shadow assassin. Although it doesn't deal a lot of damage (if you are not fed), it is still very helpful when you engage in teamfights since it gives you that important untargetbility and the helpful dash/exit.
Keep in mind that when you use your ultimate, and your other abilities are on cooldown, you can wait before you exit your target. Umbral Trespass let's your ability recharge a whole 2,5 seconds before you exit the target.
Example (shadow assassin): E through a wall into the enemy backline, do your combo (W + auto + Q. If the target didn't die by your combo you can ult them. Otherwise smite another enemy and ult them if needed. Have your ultimate for the whole duration and then your E will be back up and you have an escape tool.

If you aren't experienced with the champion, playing against a counter, or simply having a bad game, you must know how to play when you are not ahead or even. As Kayn it is easy to clear jungle camps. The most important thing to do when behind, especially against a fed enemy jungler, is to track him on the map. If you are able to do that then it should be easier to steal his jungle and most importantly not die to him. Your main goal should be to stabilize! Another key element to counter jungling is to look at the lane states. If the enemy mid and top has prio then it should be a no go.
As base Kayn you are semi strong as a dueler. Therefor when you are behind it's even worse. You should try to take easy/safe fights where you can get orbs for your form and not die. Try to play around your laners. If you have a Malzahar that has his ult up, go for the gank! It will 9/10 lead to a kill giving either him or you an advantage. Some people say inting for form is okay, but remember that you are giving the enemy gold for something you might get a minute later.
When you get your form it should be a better time for you. If behind you should almost always go Rhaast. You have a knockup and can act like a tank for your team. As an inting Shadow Assassin you'll most likely not contribute that much to your team since that form is reliant on getting an advantage.
Try to play around your team, get objectives, win fights, and win the game!! B)

Kayn is a very fun and different champion that I recommend you give a try. I have been a Kayn main since his release and still haven't got tired after playing him. This was my first guide I have ever done and it has been really fun. I hope it helped you and that you enjoy playing Kayn as much as I do.
If you found this guide useful you can always upvote it or leave a comment!

Shoutout Pontus ajajaj