Kayn has a lot of different options when it comes to early-game pathing. One thing that's important to keep in mind is that you're pretty much useless until you get your Form. This means you should try to avoid the enemy junglers as much as possible. Starting blue-side is also a possibility, but 90% of my games I follow these steps.

Step 1: Decide where you want to start. Because Kayns clear is so good he can start pretty much anywhere. Red buff, Blue buff, Chickens and wolves are all viable options. You should make your decision based on where you think the enemy jungler is pathing towards. If they are a strong early game champion you should try to avoid them. Also, consider if your laner needs the early pressure in lane. Kayn has no problem starting leashless.

Step 2: Ward the opposite side river of where you're starting. The river bush is a good place to ward. Against champions like shaco, nidalee, graves etc you should ward deeper into the river to protect against invades through the dragon/baron pit.

Step 3: Try to figure out where the enemy jungler is starting by looking at your enemy top and bot lane.

Step 4: Complete the clear. Check the next chapter for a link to a youtube video showing how to fully clear the jungle by the time scuttle spawns.

Step 5: If you know the enemy jungler started on the same side as you, and you know you lose the 1v1, E right through the wall after killing blue-buff towards the other side scuttle. It's worth the extra time to make sure you get one.

Step 6: Look for a gank. Be careful of counter-ganks.

Step 7: Go into the enemy jungle and look for camps (most likely krugs might be up).

Step 8: Look for another gank.

Step 9: Recall and clear your jungle again.

Perfecting your jungle clear is very important if you want to improve. It gives you an edge over opponent junglers that don't clear as fast as you. Youll be lvl 4 by the time scuttle spawns and should be able to pick it up for free.

Lvl 1 Single smite red-side full clear: https://www.youtube.com/watch?v=ndKvCDewQfU
Lvl 1 Single smite blue-side full clear: https://www.youtube.com/watch?v=9jppY04qxmA

Kayns weakness is his early game. He is basically not a champion until you get your form. If you're getting really fed early on as kayn, your opponents are kinda trolling. Your goals should be ganking effectively, getting orbs and power farming.

Ganking effectively: Your ganks are not that strong, the only CC you have is a hard-to-hit slow. The key to ganking effectively is to use your e to flank the enemy laner from a direction they're not expecting.

Getting orbs: A gank can still be successful without getting a kill or even a summoner spell. Orbs are the most important thing, without them, you won't become a champion. If you're not getting form between 10-15 minutes each game you need to change your playstyle. Getting it early on but at the cost of feeding is not worth it either. As Rhaast you know you can carry the late-game, so make sure you get there without being too far behind.

Power farming: Check the Jungle Clear chapter for more info. You should be having at least 7 farm per minute at 20 minutes most games. Focus on farming and getting orbs until you get your form. That's when you can start making plays for real.

Herald: Only go for herald before 14 minutes. If you can't get any plating gold for it (PLatings fall at 14 minutes) it's not worth it. Remember to always use sweeper/Control ward.

Dragons: Only go for early dragon if the enemy jungler is dead/showing top. As I've said many times, you're extremely weak early on. It's not worth risking your life for a 4% bonus AD or whatever. Just make sure the enemy aren't getting 3 dragons. That gives them a chance to get soul. By the time the second/third dragon spawns you will in most cases have form so it shouldn't be too hard to contest. Remember to always use sweeper/Control ward.

Core items: First item is always Goredrinker. If you're buying Ionian boots you can finish them before finishing Goredrinker. Mercury treads/Plated Steelcaps should be prioritized as 2nd item because you need ability haste early on to clear the jungle faster. Second item varies between Steraks Gage and Black Cleaver from game to game. If the enemy have a lot of damage you should go Steraks for survivability. If the enemy are buying a lot of armor you should go Cleaver first. Either way, you will buy the other alternative for your 4th item.

5th and 6th item: Your last 2 items will vary from game to game. Some of the options and when you want to buy each one are:

Spirit Visage: Overall good item because it enhances your healing. Buy when youre taking a lot of AP damage.

Deaths Dance: Pretty much the armor version of Spirit Visage, because its overall a really good item. Gives AD and the passive is perfect for Rhaast. Buy when youre taking a lot of AD damage.

Deaths Dance and Spirit Visage are the most commonly bought 5th and 6th items in my experience.

Chempunk Chainsword/Thornmail: Anti-heal. In most cases, Chainsword is better than Thornmail, but if you feel like you need both Spirit Visage and Anti-heal, but still need some armor too, Thornmail is the perfect option. Deciding when to buy anti-heal is based on experience and game knowledge, there is no real answer. Check the example full builds for a rough idea. Executioners calling/Bramble best can be bought earlier on in the game. The completed item should be finished last as your 6th item.

Gargoyles: If the enemy are mixed between AP and AD (plus true damage), gargoyles is a good 5th item, because it gives both armor and MR. The shield active is also good against true damage (Vayne, Kraken slayer, etc.).

Randuins Omen: Good against critical-strike champions and on-hit builds.

Abyssal Mask: Good replacement for Spirit Visage is the enemy team has one fed member that needs to be focused down in teamfights. An underrated item that was buffed recently. Secret OP maybe?

Goredrinker animation cancel: You can use Goredrinker/Ironspike Whip in between your Q dash and Q slash. To pull it off, just spam the item active while dashing and it will work
Chilling smite R activation: You can use your smite to proc your R. Good for surprise all-ins.
E marker: If you're E marker (The orb that shows your location in the wall) shows up while your cruising through your own jungle this means the enemy have vision of the wall your inside of. This can mean the area is warded or an enemy is around. This trick has saved my life countless times and helped me find wards.