There is no national database for checking criminal backgrounds, but some states do provide access to their criminal conviction databases. This site provides a list of state criminal record web sites, including information about coverage and costs of searching.
The Texas Department of Public Safety provides free access to the Texas Sex Offender Database. The Convictions Database searches require you to register for an account, and charge a fee of $3.15 for each search, and an additional 57-cent transaction fee each time you purchase credits for searches.
Criminal Records, Civil Records, Sheriff Bond Records, Sheriff Jail Records, Denton County Ten Most Wanted, Texas Ten Most Wanted, and City of Denton Sex Offenders List are available free on the Web.