Not only can you do a background check, you should do so. You can refuse to rent to someone because of bad credit or a criminal background, as long as you have firm rules and do not apply them on a case-by-case basis. This is because a disorganized approach to which tenants you accept, and which tenants you turn down, could expose you to claims of illegal discrimination.
There are many commercial services that provide background check services for you, or that make them available online. Check with other landlords or property management firms in your town for guidance regarding which resources they find most useful. Some online resources include http://consumer-guide.to/ Background.Checks and www.onsite.com. Also, do not overlook your state’s registries for sex offenders.