I'm not 100% sure, but I think that background checks include crimes, traffic violations, bankruptcies, previous employment, etc. Also, I don't think they include information back as far as you would need. Also, I do not think they include adoption information.

Have you tried getting his military records? or his birth certificate? or his Social Security application? I think these would be a better place to start. After getting more information from these, you could see what court records re: adoption are available in the area that he was likely in as a child.

Why do you think he may have been adopted?