Use "Who's" as a contracted or short form of "Who is." In other words, it represents a subject plus a verb. Some examples of correct use are: Who's my partner? Who's on third base? Who's the smartest in the class? Who's absent today? In every case, the spelling "Who's" represents the main part of the sentence. In these examples, the sentences are questions.
Use the spelling "Who's" to represent the subject and verb "to be" in a dependent clause (secondary clause) in a longer sentence. These sentences may be statements, not questions. Some examples are: I'm looking for a man who's strong and brave. She is the woman who's from Canada. In any case, "who's" still means "who is."
Use "whose" as a possessive meaning "belonging to." It always precedes (goes before) a noun. Some examples in questions are:
Whose book is this? Whose cars are those? Whose dog ran away? Notice that it can go with a singular or plural noun.
Use "whose" correctly as a relative pronoun. In sentences with two clauses, "whose" can also introduce a dependent clause. These may be statements, not questions. Here are some examples: I want a man whose car is reliable. Let's find a company whose offices are close to here. In the first example, the car belongs to the man. In the second, the offices belong to the company. In both examples, "whose" shows possession and precedes a noun.
Finally, practice the difference between subject plus verb (who's) and possession (whose) in these contrasting examples: (1) Who's that young man? Whose car is that? (2) Who's the lady in the red dress? Whose dress is that? (3) Who's the pretty lady? Whose wife is that? Practice writing some more examples on your own. You will be a better write or student when you use "who's" and "whose" correctly!