I'm a firm believer in the second amendment, but JUST having a background check? That's a bit dumb. In my opinion, you should be able to have pretty much any gun you want, so long as you pass a background check, have proper training, pay for the weapon, and abide by other such gun laws. But JUST a background check? No.
@Gun Having a psych evaluation wouldn't help. I've done quite a lot of research in regard to the topic, even having had conversations with psychologists, and the general consensus is that it wouldn't help. Sure, it might deter SOME people that have psychological issues, but that's the minority. The majority of gun related issues are caused not because someone has a psychological issue when they purchase the gun, but because they snap. Forcing people to have psychological evaluations would just end up making guns more costly for either the government or for the people purchasing the guns.
Well, the second amendment doesn't bring up assault rifles since they weren't invented until the 1890s. It says the right to bear arms shall not be infringed upon, but I don't know if that was meant to mean for militia use only or for collecting.