We have found that the only people that want to move during this difficult time of the year are people currently being kicked out of their existing living situation, people with criminal backgrounds, and occasionally we get luck and find a person changing jobs.I require my PM to sign them for about 16 months at this time of the year, so we don't look for tenants at this tough time of the year. 

You start calling and complaining and some how that info will find its way to the local criminals who you are screwing with .  

Also, what type of background check should it be, should it be a credit and criminal background check?

Usually for someone that has a caregiver living in the unit with the disabled person we would require that the company employees were screened for criminal by that company,we listed as tenant #2 on the lease as  "Authorized caregiver of XXX Company Name" as tenant #2, as a caregiver, because they usually wouldn't stay the entire day with the person, this permitted a rotation for the caregivers and worked out fine for us, as the caregiver wasn't financially responsible for the rent. 

Allowing criminals to simply walk away scot free is a unfortunate side effect of having so many unprofessional landlords.

Many times the seller of the note will provide collateral files that may have some borrower info, and we also research social media, PACER for bankruptcy, and any sources for criminal records to develop as much info that we can to underwrite a NPL. 

When landlords did not have access to credit reports or criminal background checks like they do now, then charging a half-month's rent seemed fair as they could do what you could not.  

Security deposit plus move-in fees cannot exceed the amount of the first full month’s rentNon-refundable move-in fees are limited to only tenant screening reports, criminal background checks, credit reports, and cleaning fees.Total non-refundable move-in fees are limited to 10% of the first full month’s rentPayment Plans Required –Landlords must allow an installment plan to pay a security deposit, a pet deposit, move-in fees, and last month’s rent.

Credit score disqualification, insufficient work history or income, criminal record, etc.

If the previous address is close, march into the court house and look up the criminal and civil filings under the applicant's name.