It depends on the employer, sadly it's the way things are these days. Background checks will look for any criminal activity that could be in your past, even something completely silly you could have done growing up. They also look at educational background, such as proof that you really have a high school diploma or a college degree. They also routinely run credit check looking usually for a FICO score. If you have a bankruptcy then it would worth mentoring when it comes time to do the background check. If is a game changer for an employer it's better to know up front instead of having an offer rescinded, right?
That's why I keep stressing that a half done name change is going to hurt and not help. When I was doing mine, I was sending copies of my name change to credit bureaus, creditors, utilities lIke phone cable or electric, my high school, college, everywhere I cold think of. Getting a good job, I mean one that pays, is a lot more effort than a drivers license or social security card change.
Like I said, email if you want info. I've been full time for almost 13-14 years now and have gone through a few job changes. I know the scary new full timer feeling that comes with having no employment and prospects that might look bleak.