If you are seeking a state or federal background check on YOURSELF, it should be easy. Simply apply for any of the many jobs that are usually available at your state or local police department or homeland security office. They are sure to run a background check on you (though you might not like what they find), and you can always obtain the information through state and federal Freedom of Information rules.
If you are seeking a state or federal background check on SOMEONE ELSE, especially someone not related to you, it gets trickier. There are many private investigators and security services that can find out information for you, but its completeness and accuracy is probably dependent on who you enlist, and how much you are willing to pay. Depending on someone's celebrity, and the kinds of information that's out there (divorce, arrest, child support, home ownership,etc.), you might find info via the internet and gossip sites, but be wary of its accuracy as well.