If the targeted creature is an illegal target by the time Grimgrin's last ability resolves, the entire ability doesn't resolve and none of its effects will occur. You won't put a +1/+1 counter on Grimgrin.

If the defending player controls no creatures when Grimgrin attacks, the last ability will be removed from the stack and have no effect.

If Grimgrin's last ability resolves, but the targeted creature isn't destroyed (perhaps because it regenerated or has indestructible), you'll still put a +1/+1 on Grimgrin.