Checking a tenant's credit report is a good way to hedge against tenants that might not pay rent or could cause damage to your property. Permission is always required to pull and individual's credit report.
Before running the tenant screening on the applicant, you'll want to sit down and interview them. There are basic questions that should be asked. Obviously you'll need their name and contact information if they're serious about renting from you, but other subjects that you may want to touch on are the applicant's employment and how many people will be renting the unit. Keep in mind that there are certain questions that are illegal to ask and make sure that you follow both local and federal guidelines so that no one can claim discrimination. You may certainly narrow down the search for the best possible tenant by weeding out tenants with criminal histories or poor credit scores.
Some applicants will actually be able to provide a copy of their credit report and score when they apply for the unit. If not and the landlord chooses to run a tenant screening on them, they must provide permission to do this.
Following up on references is a good practice as well. The applicant may provide a list of their previous landlords to you and their contact information so that you may speak to them and verify what kind of tenant they were previously. Keep in mind that the most recent landlord may not be 100% honest if they would like the tenant to leave.