There several viable kayn pathings right now.
First one is > > ....then you get level 3 and best way to go is by ganking midlane so tell your midlaner not to push.

Second one is Wolves> > > ...try not to die to krugs cause its highly possible if you are not carefull enough...you can clear without smite until the krugs and then use the smite on the big one when he is low...KILL THE SMALL ONE FIRST OR U DIE AND U GET FLAMED EZ Clap
Third is a cheesy pathing...get leash > >Wolves ( let the big one live) and then gank mid or bot ( top ) depends on side of map

As Kayn you have obv 2 basic item paths...1 for rhaast and 1 for SA...there are few more but they are more like troll builds in a way.
You obv wanna start with Hunters Talisman and Reffilable potion
On your first back you want to decide if you want to either clear or go gank...if you want to clear take stalkers blade and if you want to gank take 2 longswords. If you feel you need healing debuff early cause some healing champs like vlad/soraka feel free to buy executioners calling. Get control ward if you can cause its really good for ganking cause you can see where laners have vision.

As rhaast you want to go more sutain way. I would recommend going for black cleaver after Warrior but if you want u can go deaths dance...some kayn mains say that its viable aw. Boots are either Mercury vs AP/CC or Ninja tabi vs AD. Vs AP go spirit cause it gives you mr and it gives 20%+ healing which is very good on rhaast. Then i would go deaths dance and if you are really ahead you can go yoummu to snowball.At late go sterak/GA (if you have executioner go mortal reminder)

Now for the Shadow Assasin. As SA you wanna go full assasin obv. Go Warrior into Duskblade ( Duskblade is core item on SA) I found a lot of success with edge of night but you can go yoummu 2nd if you want. Then go yoummu or regards/reminder. GA is good purchase as well. As boots you want to go with mobis to snowball or sorcery boots for extra dmg if you need it.

Most asked question about Kayn is: What form i go?
Let me tell you!
Its pretty simple...if you can 1shot 3 champions from enemy team you go SA. If you cant go Rhaast. Simple as that.
BUT!!!
If you are against enemy team where you can oneshot only 1/2 but they are carries its very good to go for SA even if it means waiting. If you dont have tank and have large amount of dmg already go rhaast and same as with SA even if it means waiting for the form.
ALSO YOU SHOULD HAVE FORM BEFORE THE 15MIN MARK!!! Kayn has very good mid game powerspikes and you can 1v9 games with him...OUR ANIME BOY ;)

Thanks for checking out my kayn guide.
Hope u will have fun playing Kayn cause he is PogChamp

And remember: Enemy team Vayne---->SA as mr.Karasmai said :)