Rush Stalker's Blade, then boots, then warrior, then Black Cleaver. Then get either ninja tabi or merc treads depending on enemy team. Typically your next item would be a Spirit visage, if you get that you'll never die in fights especially team fights. But it'll probably be last item if there is lots of AD w/ none or only 1 ap. But if that AP is carrying rush it hella. Your last two items are game dependent. The best two items would be a Death's Dance & guardian Angel(even though you already capped your CDR), DD lets you be able to get more abilities off and heal more in fights which means youre even less likely to die.

Other final two items if necessary:
Adaptive Helm (again the CDR is sad but if you find yourself getting to 60% cdr often you can take sorc as your second rune page and get Transcdence which is quite strong for red kayn, but means you lose increased healing from the resolve tree)
Edge of night if you are the super hard carry
Ohmwrecker has usefullness for late game diving, but is not ideal in most cases.
Randuin's Omen is a godsend against hard carry adcs who are kiting you
Righteous Glory can also be useful if you find the enemy team keep getting away just barely after a quick poke engage (better for higher elo)
Sterak's Gauge is a good item for when you are tanking everything and doing the damage for your team (high recommend last or second to last item for low ELO games)
Titanic Hydra is a great item for Kayn when he is carrying. Especially if you build a tiamat early then it increases your clear speed by a lot and you can supsequently kill enemy laners a lot easier with additional damage potential. However, it doesn't scale well with Kayn (especially red) into late game as it is kind of a jack of all trades item, and late game your clear is perfect without it killing most camps quickly with your aoe spells w/ 40% cdr.
Gargoyle Stoneplate is another item that is super useful when the enemy isn't building Magic Pen or armor Pen and you're the singular tank. Additionally the active is insane, and the passive is insane in team fights. Make sure to be near 3 enemies when you pop it, you double your health and the item will now be giving you 80 armor and 80 MR. But again if you're the only reliable person on your team its not great because that extra hp means you're not doing damage, so if you go gargoyle make sure you and the support are able to protect the ADC so they can just kill the heck out of the enemy.
Locket of the Iron Solari: This item is good for kayn especially if your other late game item is an HP item. At level 14 you give a flat 240 hp, plus 20% of your bonus health shield. This shield is also nice if you're behind because the flat shield uses either your level or the targets, whichever is higher. SO if you been feeding all game and are level 12 and your adc is level 14 (god forbid) and you pop the shield instead of giving 210 shield you give 240 (plus % bonus hp). All around a great item for a tank who is behind. But this item will more than likely HALT any snowballing. So if you been carrying your game and buy this you'll realize you didn't get a powerspike and feel behind. This item is great for team fights and for helping support your team. So get this if you're behind.
Zz'Rot Portal: so the last item to consider is zzrot, this is an item that is good for games where your team won't group and siege properly and you just need to get more damage output on objs. You can't be everywhere but with zzrot you kinda can... drop a zz rot bot then shove mid enemy team especially in lower elo will have no clue what is going on. WIth kayn's E you can actually sneak the zzrot into a lane and they might not even realize you've done that. It also helps you get the last bit of damage on a turret that has just enough hp you can't dive. The voidspawns (annoying little buggers that kami kaze the towers) do damage based on your maximum health. SO again... GET MORE HP BB.

The last item decision you have to make is late game and I mean very late game if you sell stalker's blade for a different item. Only do this if you already have plenty of DH stacks. Because I cant tell you the # of times I have gotten a few stacks from a blue smite, even killing people with like 500hp with a single smite cause of DH pops. You can also sell boots but if you do that you just got to be careful because that means you can't chase down enemy adc's who are kiting or reacting to a teammate getting caught out near you as fast. But with your e if you use it right then it should be fine.