It costs about $5 to run a public background check in Texas. I don't really think it would be a huge financial burden for private sales.
Of course a lot of these proposals are ignoring how the crazies acquired guns in the first place. We should definitely have a more comprehensive mental health system that coincides with public education. From there you have to make the hard decision as to whether it's legal to abridge someone's right to own guns if they pose a serious risk to others due to violent tendencies.