When you say background check, do you mean credit, criminal, prior employment...what?
If it was a credit check, the bureaus are required to keep a log of all inquiries and disclose those to you when you pull your reports. Some states allow you one free per year, otherwise it is a small fee (usually about ten bucks per bureau) and you always get a free one if your credit report was the reason for not getting a job, apartment, car, etc.
But as to the other records, I haven't the faintest... Good luck!