In Texas and North Carolina, all you need is reasonable suspicion that your tenant is involved in drugs and you can get rid of that tenant right away. In New Jersey there has to be a criminal conviction before you can evict a tenant. Many states, however, now make it much easier for landlords to get evictions based on illegal drugs.
Drug evictions start off with the Unconditional Notice to Quit. In most states, if the tenant doesn't leave after receiving the notice you can then file for an eviction and have a hearing within a couple of days.
Simply finding out that a tenant has prior criminal convictions does not give you the right to get rid of a tenant who is in compliance with all of the lease provisions. The only way to end that tenancy would be if you discover false statements or information on the lease.