If my boyfriend has stayed in my apartment for the past 3 months but I’ve asked him to leave yet he refuses, what is the appropriate legal action to take at this point?

You have a couple of different options depending on the extent of your situation and the cooperation of law enforcement.

A simple remedy is for you or the landlord to serve a criminal trespass warning on your boyfriend.  This is the cheapest and quickest option, but it depends on the cooperation of your local police.  Some will follow-through and trespass him-- and then arrest him when he violates the warning.  Some agencies, however, will not do the criminal trespass because he has obtained the status of a tenant.

Your second option is eviction.  Either you or your landlord will have to file eviction proceedings against your boyfriend since he is not an authorized occupant.  The State Bar of Texas website has some good resources for eviction procedures that you may want to look at.  The same handbook discusses your duties for the return of property in an eviction case.

If this has been an abusive relationship, then you may have a third option-- which is seeking a protective order.  Under the Texas Family Code and Code of Criminal Procedure, judges are authorized to grant protective orders which exclude abusers from the residence of victims.  This type of protective order is called a "kick out" by some district attorney's offices.  The procedures tend to vary slightly by county, but many district attorneys offices will publish their procedures online-- so check out your county's DA's website.  If they don't have them online, make an appointment with the DA's victim assistance coordinator to see what the procedures are in your jurisdiction.  When the judge issues the order, you can also ask for the judge to impose a set time for a friend or relative of your boyfriend to come and pick up his things.  If he fails to do so, then the property can be considered abandoned in the protective order context.

Assuming that, in addition to not being on the lease, he is also not contributing to the rent, he is an invitee or guest. A guest may be asked to leave at any time, without prior notice; if he does not, as he seems to not be leaving, the next step is to file an "ejectment" action against him. Ejectment is essentially eviction for non-tenants; it is somewhat more  formal in its requirements and complex than the typical eviction action, so you would be well-advised to have a landlord-tenant attorney help you. The action can also require him to remove his belongings from your storage locker.