In most cases they are just going to check to see if you have voilent crimes. A big one they look for is domestic violence charges, but a lot of felonies can prohibit you from owning a firearm.
You would know if you couldn't own one, they probably would have told you in court when you were sentenced.
I honestly couldn't tell you how well the background check is though.
If you have never had any legal problems or been considered mentally incompetent (sp?) I wouldn't worry about it.
I can't imagine them doing a credit check unless you were buying a very expensive rifle and financing it. I have put guns on law a way before and they still didn't do a credit check. If credit was an issue I wouldn't own any guns at all

.
It is going to depend on what state you live in though. Some states require you to have a firearms card a FOID card I think it is called. And if you decide to get a handgun it is a bit different in some places.
Paul