Who will be doing this checking, and what are they checking for?
Will your voting record be checked?
What is required to pass this check, and were can I find the standards?
Will anyone that is denied arming themself have any recourse?
How is going to pay for this background check?
Then once the background check is done, where are the results stored and for how long?
See there are a lot things to think about.
How about anyone wanting to buy a gun must not be in a database of felons? That's it! Not a felon no reason to "infringe".