You can run a background check on anyone.
Most background check companies collect and publish information that is publicly available, anyway. Criminal records and court records are public information that was created at tax payer expense. However, you would have to visit every state, county, and city to be sure that you covered everywhere that your subject person has been, and where she might have committed a crime. That is why background check companies are so useful, because they gather the information nationwide.
The Fair Credit Reporting Act (FCRA) requires employers, landlords, and financial institutions to get a person's permission to run a background check. So, they just have you agree to a background or credit check when you fill out their application.
However, a private individual is not affected by the FCRA, and you can gather information on anyone from any publicly available source.
You can find out how to do a free background check, here: