This is another one of those problems of mistaking an apostrophe for a possessive. It doesn’t sound likely, I know, but it really is an easy mistake to make. It happens with its/it’s, and it happens with whose/who’s. (I’ve even seen it with hers/her’s, and her’s isn’t even a word!)
Whose, the word without the apostrophe, is the true possessive pronoun; think of Robert Frost’s poem Stopping by Woods on a Snowy Evening: Whose woods these are I think I know (the woods that belong to someone).
Or for a more modern-day illustration: Whose button popped off in the dryer?
Who’s is a contraction for who is or who has. Let he who’s (who is) without sin cast the first stone. Is he the one who’s without sin? Are you the one who’s missing a button?
If you can substitute who is, then use who’s; otherwise, use whose.