Ask the person to allow you to request a criminal history search of their background. Those that don't have anything to hide or are not strictly private will not have much problem with this request if it is for a legitimate reason such as employment, residence applications or other legitimate requests. An FBI request for a criminal history can be performed without much effort by visiting the FBI website. A short form is available to complete to receive the arrest history (see Resources).
Secure every piece if information you can find out the person who's arrest record is being checked. Information such as their full name, a current address, where the person has lived in the past, the drivers license number (when possible), their date of birth, hobbies and the names of those whom the person may have been married to or in a relationship with.
Write down any relevant court cases connected to the person your reviewing. Be sure to review the case witnesses and co-defendants to check for other names that may match your person's other criminal cases.
Go online to the court website or sheriff department inmate look up where the person may have previously lived or worked and enter any identifying information into the database to locate a match. This can yield the arrest history and also point you to other counties to search in if the person had a hold for other counties due to other criminal charges.