Can you become a mercenary in Skyrim?

Not really.

You can do small missions to take care of people, but there are no mercenary missions. Atleast none where you gain that title.

the most way to be a mercenary is to go to any inn you can find and ask for work they should give you a quest to kill a random bandit leader or some random quest that you can do for gold this is about the closest to being a Merc you can get in Skyrim