Of course, there's an excellent argument to be made that violent felons should be prevented from owning a gun. However, background checks are inherently unjust toward honest citizens as they impose prior restraint of a Constitutional liberty. Any law preventing violent criminals from owning a gun should provide punishment for violating the law, not background checks on everyone. In addition, many juvenile crimes do not warrant denial of a firearm. Someone who gets caught with drugs as a teenager shouldn't be prevented from owning a gun as an adult.
Many people can reform from even a violent crime committed as a child.
About the only law everyone agrees on, in principle, is one banning straw purchases. That is, it should be illegal to knowingly purchase a gun for someone who cannot legally own one. The principle is that it's wrong to help someone else commit a crime. The person who drives the getaway car should face reprisals along with the bank robber. The problems come when the law arbitrarily delimits who can own a gun.