For Arizona background check resources that involve family relationships, you can get many kinds of records from the Clerk of the Superior Court. They have offices in Phoenix, Mesa and Surprise but can also be reached by phone or fax. They will issues copies of marriage and divorce records, as well as some civil court documents. There are fees associated with any request and it can take up to 2 weeks to get your copies.
The Arizona background check resource that is involved in maintaining criminal history records is the Department of Public Safety (the Criminal History Records Section, to be more precise). This material is only handed out to authorized organizations for the purpose of conducting employee background checks. You can also get a free copy of your own criminal history if you need to see the documents.