If it's too complex or will make me look like some kind of criminal, I can simply not do it or do it for free (since money is usually what complicates things anyway).

I believe there are five levels to choose from - basic through national criminal check and eviction report

—In all criminal cases, contempt cases, and other cases filed pursuant to this chapter, if a party has sold, leased, or let real estate, the title to which was not in the party when it was offered for sale, lease, or letting, or such party has maintained an office bearing signs that real estate is for sale, lease, or rental thereat, or has advertised real estate for sale, lease, or rental, generally, or describing property, the title to which was not in such party at the time, it shall be a presumption that such party was acting or attempting to act as a real estate broker, and the burden of proof shall be upon him or her to show that he or she was not acting or attempting to act as a broker or sales associate.

I screened a tenant in which a criminal background came back in a state where he's never claimed he's been.  

., @Dovid Staples, @jeanette alder,  So I emailed my lawyer and he responded, "He actually has some criminal liability as I think the tenant could make the case for theft, but best to get it settled and if you are happy with $800 that is what matters."

You need your own background, income, credit, and criminal check in place.

I do anyway, but the county doesn't allow people on the program if they've got criminal or eviction history.3.  

I hate that kind of governmental oversight which assumes I basically am a criminal and have to explain myself to prove that I am not.Also, how will this affect closing with the seller?

I highly recommend doing a credit and national criminal background check on all tenants.

—In all criminal cases, contempt cases, and other cases filed pursuant to this chapter, if a party has sold, leased, or let real estate, the title to which was not in the party when it was offered for sale, lease, or letting, or such party has maintained an office bearing signs that real estate is for sale, lease, or rental thereat, or has advertised real estate for sale, lease, or rental, generally, or describing property, the title to which was not in such party at the time, it shall be a presumption that such party was acting or attempting to act as a real estate broker, and the burden of proof shall be upon him or her to show that he or she was not acting or attempting to act as a broker or sales associate.