Not "anyone" can purchase a firearm legally. All individuals who go to a legitimate FFL dealer, have to fill in the background check form. "Straw purchasers," people that don't have a felony background and buy for others, have been a problem, however, the government is changing that now.
The only ways to obtain a firearm and avoid the background check are:
1. Go to your preferred inner-city drug dealer or other criminal and find out where to get one. They'll point you to someone that will sell you a stolen firearm.
2. Find out who owns firearms, wait until they leave home, break in and steal them.
3. Find out who makes "ghost-guns" and purchase one.
Bottom line, as I said previously on this, if someone can't legally purchase a firearm, that individual is someone that shouldn't have one, as that individual will no doubt commit crimes with one.