How to do a background check?

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Mark,
All varies by states.
Possible privacy issues if not authorized by the person who it is being run on.
Possible issues for anyone doing you a favor and running this stuff as could lose their job.
I deal with backgrounds extensively as part of managing a fraud investigation operation for numerous states.
Best advice I can give you, if you want to save time of doing all the legwork yourself, Google it and see how much it will cost you and how much can you legally secure in your state via a PI or vendor that specializes in this type of data collection.
Doing it yourself is tough. As an example, I recall when I bought my condo in Miami for my mother in law. They wanted a background done on me via the Sheriff's department for the county I reside in. Well, in Florida, when I got the background for them, it was for Palm Beach, where I reside. The condo is in Miami. If I would have had a record in Miami or any other county, they would not have known. They would have needed a report from each and every county in Florida. Whereas in contrast, there is a Florida website that does in fact provide all that. Just a matter of knowing where to go for your data.