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The best thing to do when playing as Greystone against Khaimera is first observe his behavior. Khaimera is popular with newer players, so you might've gotten lucky and are now playing against one. If this is the case, he might be unaware of how Reforged works. Even if he does, some Khaimera players will decide to commit to chasing you up your lane and trying to score a kill. Bait him a little to make sure he will be too far from his tower to get away, and then engage him with Make Way and capitalize on the fact that Khaimera has no escape mechanics. If, for some reason, he manages to put enough distance between the two of you to get out of your melee range, immediately follow up with Assault the Gates. Remember that Khaimera is very fragile and his only survival mechanic is Spirit Regeneration, which means he might decide to turn and start fighting you to gain back some health or dissuade you from pursuing. This should make killing him easier as he is more likely to back up into a wall.
Also watch for Khaimera's mana, as he can easily exhaust himself by using Unleash to jungle and clear creep waves. This can mean an easy kill on the defenseless little kitty :)