Yes, you can legally apply for a name change. The procedures are complex. The real basis for denial of a name change application is if the court finds there is intended fraud involved. As long as there is no basis for finding fraud, the prior conviction should not iterfere.
You can legally change your name but that will not erase your past. It it could every felon in CA would change their name. Your record might read after a name change "Old Name, also known as New Name" or vice versa. You can't escape the past.