The difference between the use of whose and who’s is similar to that between the use of its and it’s (discussed here).
whose – the possessive form of whom.
“The director, whose shares were recently acquired by the company, resigned last week.”
who’s – a contraction of who is. When you see the apostrophe, think “who is”.
“The plaintiff, who’s suing the defendant, is represented by a very competent counsel.”