This one can be confusing for the same reason that it’s and its can be confusing. That blasted apostrophe in “who” can fool us into thinking it’s a possessive. But it’s a contraction. It combines who and is.
So if you can say, “Who is (whatever),” then you’re OK with using who’s. “Who’s coming over?” and “Who is coming over?” mean the same thing.
But whose means belonging to the person you’re referring to. “Whose coat is this”? is accurate. “Who’s coat is this?” ends up meaning “Who is coat this is?” That’s not cool.
The possessive is whose. Who’s is a contraction of who and is.