People convicted of a crime involving violence and those adjudicated as incompetent are already prohibited from possessing firearms according to federal law. The standard background check should and quite often does catch those things. Some of the other prohibited persons are harder to catch like those who use or are addicted to controlled substances, fugitives from justice, etc. As is continually illustrated though, crooks are good at getting guns regardless of the law.

http://www.atf.gov/firearms/how-to/identify-prohibited-persons.html

We are never going to get all of the nuts in a screening process, but I am all for more stringent filters. They are an inconvenience for honest and responsible folks, but they will save lives. After the initial screening, you should be good to go until you do or say something stupid, like our boy in the video there. Once you have demonstrated your lack of worthiness or responsibility, privileges should be lost. Who gets to define stupid and what the revocation of privileges and reclamation of firearms involves seems like pretty tricky if not constitutionally invasive territory.

I am not necessarily against required registration and would even support some sort of competency/proficiency test prior to granting ownership. Again folks with good intentions really don't have much to fear other than more paperwork and time, neither of which is really a big deal in the end if you are buying a gun for the "right" reasons.