In the U.S. there is no way to do it. There is no federal registration, and most states do not have registration, and in the few that do, it's very unlikely that you would be able to access the information in the first place.
In the US, it is illegal for some people to own a gun- convicted felons, persons that are mentally incompetent, etc. The background check is to insure that you are not a prohibited person.
In the US, to "engage in the business" of buying and selling guns, you need a Federal Firearms License. Those are issued by the Federal BATFE, and DO require a background check. Several states also...
When you purchase a firearm from a dealer, they phone your information in to NICS. NICS conducts a background check on you, and determines if the sale will proceed or be declined.
Some are easier to research than others. For example there is a Remington Collectors Society, but you cant find much information on Remington serial numbers. On the other hand if you have a...