This is just my idea, so people can feel free to pick it apart. Require all gun sales to be performed by a licensed gun dealer, or if it's a sale between private citizens, arranged through the local police station or sherriff's department. The dealer or law enforcement will perform a background check on all purchasers.
Anyone charged with a violent crime (assault, battery, making threats) should have their rights to purchase, carry, or own guns suspended. Offenders can have their rights reinstated by completing an anger management course and getting letters from two people who know them well and are willing to "sponsor" them. If that person goes on to commit a crime with a gun, their sponsors will also face criminal charges of some kind. That puts skin in the game for both sides - don't sponsor people you don't really trust, and don't commit crimes because it will land your loved ones in jail.