Generally, you can run a criminal background check on anyone using public records without the person's knowledge. Criminal records are public records and accessible to anyone. Many internet sites exist, including www.publicdata.com, which will provide you with the information for a minimal fee. I would make the applicant aware of your intention on the application and get the applicant's permission. For example, the application should state, "by signing below, you consent [business name] to run a criminal background check on you".

As for a credit check, I would question the necessity of the credit check and its connection to the position. There might be positions that a credit check is necessary; however, without that knowledge, I would steer a potential employer away from any such check. Of course, if the applicant consents to the credit check, then the potential employer should be able to do so.