You have to evict him in total, You start by reading the statutes which I presume will require service of a Notice to Quit before serving the complaint. You probably should name both the dwelling (apt) number and the storage unit. As he did pay rent without a written lease, he is defined as a month to month tenant.
I try to start my answers the same way - reminding the reader that this advice is worth what you paid for it! Also, I recommend a consultation with a knowledgeable attorney because the facts of your situation would likely affect how the law applies to your case. That said, I will answer the questions assuming that you own a storage unit, rented it to a friend who has property in the unit and had access to the unit, and he has not paid you rent or has fallen behind on his rent related to the unit.
You actually raised an interesting question. Your normal Landlord/Tenant relationship is governed by the Landlord/Tenant Act. It provides Tenants with a lot of procedural and substantive protections. The reason for the protections lie in our fundamental rights related to the place we call home.
Storage units, however, are simply places where we store stuff and/or junk. As a result, the same level of protections for the the Tenant (or, now more correctly referred to as the "Occupant"). In fact, storage units are not even covered by the Landlord/Tenant Act. They are covered by their own statute the "Self-Service Storage Facilities" Act (73 Pa.C.S. 1901) [http://www.selfstorages.com/lien_law/pa.htm].
The procedure under the statute for recovering rent from a defaulting occupant is the basis for television shows like "Storage Wars" on A&E. In essence, you are required to provide the occupant notice of his default on the storage unit in conformity with the requirements of 1901. Then, once the appropriate period has expired, you must publish (again, in conformity with the requirements of the statute) notice of an auction of the property in the unit. Then, if the occupant has not satisfied the monies owed and the requirements of the statute were followed, the property may be sold at auction and the proceeds kept to satisfy the balance owed. The excess amount, however, must be held for a period for the occupant and then, if unclaimed, turned over to the government.
For more guidance, review the statute and consult with an attorney. An attorney can help ensure that the requirements of 1901 are satisfied, which paves the way for you to recover both the money owed to you and the ability to release the storage unit. Hope that helps!