You could without a background check, but you would have to either go through the gun show loop hole (basically states that have gun shows that do not require a background check to do so). Also, certain private sellers will also sell you firearms (I don't know any sources) so you would need to do some search on that yourself. I cannot advise you further details without going into the territory of illegality and also keep in mind of your state's laws as well as federal law, both for the buyer (you) and the seller (private party). Some states require background checks for ALL firearm transactions, but all FFL (Federal Firearms License) dealers will require a background check. Also, keep in mind that if you are purchasing from a private seller, if the private seller has any suspicion that you may be a prohibited person (felon/adjudicated mentally defective/domestic violence/etc.), he/she may refuse to sell to you (it's to protect themselves from serious legal trouble).