MC, Kly made a good recommendation. But, I'd be wary of using an online site because you'd establish a record for yourself as someone who was looking into the possible criminal background of two people. You'd have no control over how the site's operators would use that info on you.

Do you know someone you can trust in law enforcement? You could ask for a referral to a private investigator or someone who does background checks on, say, day care workers or religious school teachers. That way, you could pay the background checker without having to give any personal info.