"Can you check Mr. A's profile to see if he's related to Mrs. B?" sounds more natural than what you had written. Outside of the context of an investigative setting, a profile is someone's personal information (name, address, date of birth, email address, phone, etc.), like when you sign up with a new website or get on a mailing list. A background check infers checking for any criminal history or legal problems, or maybe past names used. In the context of an investigative setting, as you have presented, "profile" is better to use than "background check"; a "profile" in an investigative situation would include the personal information as well as the criminal history of someone. A "background check" is something typically performed by a potential employer, landlord, etc. rather than by the police.