What are my rights when kicking a roommate out who is not on the lease?

Since this roommate paid rent (I assume at some point) but is not on your lease, they will be considered to be your unofficial tenant or "subtenant" (and you are the "sublandlord" or "master tenant"). Since you have the legal right to occupy the premises, you can file an unlawful detainer action (i.e. eviction proceeding). This means however that you will have to comply with all legal requirements in order to get this person removed. This starts with giving your sub-tenant written notice (30 days for a month-to-month tenancy of less than 1 year; 3 days for non-payment of rent). You will then file suit in court. If successful you will be issued a writ of possession. Your roommate/subtenant will either have to leave the premises by the specified date or you can have the sheriff remove them. 

In the meantime do not undertake any "self-help" measures such as changing the locks, removing their belongings, etc. You could be sued if you do. What you can do now is to contact a tenant's right group or attorney who specializes landlord-tenant cases; they can advise you on the correct way in which to go about all of this.

Note:  If this roommate is an "official" tenant, the only way to have them legally removed from the premises is to have your landlord file for the unlawful detainer action. Having your roommate's name on a lease is not the only way that they may be considered to be a formal tenant of your landlord.  In addition to being on the lease, they may have achieved the status of a tenant if your landlord accepted rent from them directly. Also, if the landlord put (or allowed them to put) their name on the mailbox/doorbell, and/or if they rented the place together with you and it was clear that you were both on equal footing.