Getting Vital Records
You can check on someone's past marriage history by contacting the Vital Records office in Lansing. You an order a copy of anyone's marriage or divorce records for a $26 searching fee, which you do not get back if no records are found. If you wanted to access birth records, you will have to be a family member, or wait until it's been 100 years since the certificate was issued.
Another Michigan background check resource is the Criminal Justice Information Center (CJIC). They maintain a database of all state criminal records and these are public records to be accessed by anyone. You can print out the forms from the CJIC website but you have to mail in a request to get any information back. Each search will cost you $10, regardless of whether any records are found.