When Kayn is in terrain near you, you will get a visual warning on the edge of the terrain he's in.
Damage Kayn while he's in Shadow Step to shorten its duration, or apply hard crowd control (stun, charm, knockup, etc) to end Shadow Step immediately.
Thunderlord's Decree is the go-to when going for the Shadow Assassin form, as it is on any assassin, since the extra damage is needed to ensure that he can burst down squishies. However, Stormraider's Surge is a viable alternative that, while reducing his damage potential, makes him less prone to kiting. It is a especially good choice against champions that rely on slows to kite Kayn, examples include Ashe.
When going for Rhaast, Fervor of Battle is the superior choice, since he will be more focused on dealing sustained damage.
Grasp of the Undying is an alternative that gives Rhaast more sustain and grants him the ability to use Swiftness, which gives him a tenacity (this is a big deal if Rhaast is using Ninja Tabi as he will still have tenacity). It reduces Rhaast's damage output, however.
Duskblade of Draktharr works well with Kayn's Shadow Assassin form. With his ult, Kayn becomes unseen by the enemy, allowing him to potentially proc Nightstalker a second time.
Because Rhaast is reliant on healing to last during teamfights and to duel enemies, it is recommended to buy Grievous Wounds items to have a better chance of fighting him. Keep in mind, however, that Bramble Vest and by extension, Thornmail, are less effective against Rhaast than other healing reliant champions. This is because he heals with his abilities instead of his basic attacks.