The main Nevada background check resource for any sort of criminal history is the state Department of Public Safety but these records are not freely open to the public. If you have a signed permission statement from the individual, then you can make a request to get their full criminal record. You'll have to pay $21 for each person's history and it will include arrests, convictions, indictments, acquittals and more). They do not offer online services for this and it can take up to 2 months to get your files back if you send by mail.
Other Record Sources
There are additional Nevada background check resources, particularly if you are looking for more family-oriented information about someone. Marriage and divorce records are held by county clerk's offices where the record was registered, and you'll have to contact the office directly to order any copies. There is a typical $10 fee for each search you do.