A lot of states have methods to remove those kinds of convictions...in some states teh removal of some is automatic after specified time periods. Some states you will need a lawyer, in WA it is set up so you can do the appeal yourself.
Check your state law. You will probably have to obtain a certified copy of the conviction to know exactly what needs to be done. Different courses for different convictions. Something like weed is probably very simple. (at least it is here)