PA-28 instructor pilot reported an engine malfunction during climb on a training flight in visual conditions. The flight returned to the departure airport; and landed safely.

Narrative

During an Instructional Flight for an aircraft checkout a ruff running engine was encountered. On runup the left mag was fouled but was cleaned out by leaning the mixture. On takeoff to 1500 msl feet the engine ran very smoothly. Five minutes into the flight at 1500 msl flying up to 2500 msl the engine started to fun ruff and vibrate. We changed the rpm's from 2500 to 2300 to 2200 and back to 2500 with the engine still vibrating and running ruff. The student asked ZZZ approach to return to airport for landing. When asked if there was a problem the student said that the engine was running ruff but no need to [request assistance] and he want to land as precaution. When returning to the airport we performed the checklist for engine loss and put the carb heat on; gas to fullest tank; fuel pump on; mixture rich; primer in and locked; megs on both; etc. Engine continued to run ruff. On final we put the prop full forward and brought the power back to idle. The engine started to cough so we increase the power slightly. We landed without incident and taxied back to parking. When taxing to parking the engine continued to run slightly ruff.