Request a Rental Application and Screening Reports. To start the screening process, send a rental application request to all interested renters.Many landlords simply fax your rental application to a screening service, then wait for the screening report to come back. When you pay for your background check and credit report on your application, they check your eviction records with the state. The tenant background check reports may include a criminal record search, eviction history information, income verification, and credit report. You then select a tenant who is qualified, eligible, and acceptable to you. Most apartment complexes will not rent to individuals with felony convictions and past criminal history involving domestic violence as well as property damages.