Providing aircraft
Let me clarify. It doesn't matter where you got the airplane. You could have borrowed it, stole it, purchased it, or rented it. As long as you are providing the aircraft and are charging for your services, you are committing an unlawful operation pursuant to 14 CFR 135. If someone else provided the aircraft and hired you to fly it, that is legal.
For instruction purposes, the student must rent the aircraft. The student is paying you for your services. Of course, the aircraft in question has to have an annual and 100-hour inspection (or an approved progressive maintenance program) for it to be used for flight instruction for hire.