There is no NATIONAL public access to such records. Some states might have public access. Many will sell the data to people like the private background check people you mention.
If you are talking about your OWN record, you can get your record from the NDR by making a privacy act request:
You are entitled, under the provisions of the Privacy Act, to request a file search to see if you have a record on the NDR. As a private citizen, you must send a notarized letter commonly referred to as a “privacy act request” to the NDR indicating that you would like an NDR file check. Individuals should send their request to the National Driver Register, 1200 New Jersey Avenue, S.E., Washington, D.C. 20590. Please include in your request your full legal name, DOB, State and Driver License Number, Sex, Height, Weight, and Eye Color (your social security number is optional). There is no charge for this service.
This is a general right you have for almost any FEDERAL file on you.