As Kayn, you should always start Red, since your clear for the raptors is unbelievably good and fast, so you want that camp the sooner the better, and since your damage is physical, it's best to keep the smite for the Blue. So Red, raptors, Blue is your main rout in most games. If the game seems like powerfarming will be the best, take wolves as well before blue and clear your jungle. You have very low CC and your damage without items is not the greatest, but you scale really well with the game so get your farming going and secure vision for your lanes.

Start with Talisman as you'll need the mana for the clear and your AoE abilities garantee a nice heal off Raptors and Kruggs. I opt for the Sabre instead of the Blade for the jungle item seeing as the most damage possible, but if you'd prefer to gank more, then please, Blade all the way as you have very low CC. Next i opt for Tabi as the choice for boots as the most trouble i had was against AD Assassins such as Yasuo and Kha'Zix, so the early armor gives a nice boost in dueling. Black Cleaver makes for a good item for him, granting the life he needs to combat and the CCR for his mobility and damage skills, same reason i take Death's Dance as the third item, since without the darkin form, he has very low healing capabilities and it makes it harder for prolonged trades. Choose between the finisher items as you see fit in the match you're playing.

As Kayn your main goal is to get farming for your items, they grant you a really good boost in damage and survivability, and securing vision for your team, so focus your early game in that.
As Darkin you're the bruiser of bruisers, something like a Fiora: High mobility, good endurance and a ton of heal. Use your E to look for flanks to get to their backline and use your ultimate if the situation gets too sticky for you to handle, it gives a lot of time for back up and the heal you get from it can put you back in the fight. If what you need is the heal to get back, you should ult the guy with the most life, since it will heal you more.
As Shadow assassin you're closer to Zed: a high damage with good mobility assassin with a nice ult for execution and evasion. Flank, flank, flank, resumes your behavior in a fight. You can use your E to move across the map very fast and it's good for rotation, so if you plan to splitpush in the game, he should be your choice.