Pathing On Kayn:
Overall, kayn has a very strong clear. He can:
1. Go raptors, E into Red Buff (smite here), then full clear blue side by going wolves, then blue and gromp simultaneously (smite here). For most people, this clear lands you at the time scuttle spawns.
2. Leash Red, E into krugs (smite the big one here) E into raptors, hit 3 then full clear blue side and take scuttle.
3. If you need to early gank bot for whatever reason, you can solo or leash red buff then get W and auto-w-q-auto-auto-q for a faster raptors to go take krugs (Q through wall and also smite the big one) then E through tri-bush or alcove for the early gank.

After this, just take whatever camps you want, if you are Red or Blue, raptos will always be your fastest camp so if you cared the jungler off, you can grab raptors before they come back. Just don't overextend.

What Most People Say:
If You watch youtube guides or read a lot of other mobafire guides, most say to play early game very carefully, to sit back and farm until you hit level 6, go for the easy ganks, etc. Most people say not to invade because Kayn is too weak early. However, if you are talking about a 4-5 man invade, it's not just kayn's early game strength that matters. So with a strong early champ on your team, some CC and a little bit of luck, you can creep behind to the enemy buff and hopefully get a pick, the buff or some summs dropped. In most cases, you'll get orbs at least, but this invade is only possible with a strong early game comp, so make good decisions. Otherwise, if you dont give the enemy jungler an early lead, then you should just path correctly and gank when the enemy is overstaying or overextended.

The best way to gank as kayn is to use your E and cath the enemy off gaurd, for example if your top laner TPs into bush while you E through the alcove to gank the top laner while they are attacking your turret, that's a perfect gank. However, you need to communicate so that you aren't just Eing in to give a free kill, and make sure that-especially when you gank bot-that you coordinate with your allies' abilities. If you do this, you should be able to rack up some kills, or orbs at the very least, which will contribute to your powerspike. Once you get form, as Red, you need to look for teamfights and objectives, but as blue, you need to look for picks on champions without their allies, and THEN grab objectives like dragon.

Mid game is when you spike. This is when Kayn dominates. You should have your form and be working on a second item, which contributes to one of the strongest midgame champs in league. At this point, when you are farming and someone is making a call, you should always prioritize the call. Drop what you are doing. Don't be the blue kayn sittting at krugs waiting for W while your bot lane gets ganked, counter gank and turn it around. At this point, you want to be pushing for towers, helping teammates get plates and keeping an eye on Rift Herald, because if you don't play really well, most Kayns fall off late game.
The Rule for building items mid game is this: If you are fed, build whatever will help you do your job the best, don't bother compromising your build for a little bit of MR or Armor. If you are behind or your lanes are behind, you need to itemize so that when you gank you can be as successful as possible. For instance extra damage won't matter against a fed senna if when you come to gank she roots you, hits you for all your health and walks away, but if you build a spell shield then you can burst her down.

In late game, your power doesn't necessarily decrease, but the enemy starts itemizing against you, you can't pop the ADC anymore unless you're full build and fed, so you start to fall off, to become less useful. While Rhaast is still somewhat useful because he can engage and just thrives in teamfights, he still will be outputting less damage and you may find yourself feeling unhelpful. So regardless of what form you are, play it safe, looking for picks if and only if they are out of position or you have a distinct advantage such as levels or number of people. Then, THEN you can make plays around objectives and push to end. In general, the late game on Kayn still feels better than early, but you have to make the right decisions and not be the one that messes up.
'Don't outplay the enemy team, wait for them to outplay themselves'
-Rossboomsocks, pHD

Kayn's a really fun champ who can be a niche pick or a monster teamfight bruiser, but either way hes extremely versatile. Just make it to mid game and you should be fine. I hope you enjoyed the guide and that it helped in any way.