PA-28 Pilot reported erratic airspeed indications and engine roughness shortly after takeoff. Pilot returned to departure airport for maintenance.

Narrative

At ZZZ XA:00 pm local time on Date I returned to land due to a static port clog as well as engine roughness. After getting cleared to takeoff; I departed Runway XX with instructions to climb and maintain 3000 feet and make a right turn to a heading of 290. Everything was normal on the takeoff roll. On the initial climb; I noticed that my indicated airspeed wasn't increasing past 65 mph on the initial climb. I immediately lowered the nose to avoid a stall and monitored the engine instruments. The engine instruments were normal; but my indicated airspeed wouldn't increase past 65 mph even after lowering the nose. This indicated to me that I had a clogged static port. I informed the Tower that I needed to return to land on Runway XX. He told me to report midfield downwind Runway XX. While turning around to enter downwind for XX; I experienced engine roughness and vibrations. While making a right turn back to enter the downwind leg of XX; I lost sight of the runway. I had to make another 360 degree turn in order to find the runway and enter into the downwind leg for Runway XX. To confirm my suspicion of a static port clog; my indicated airspeed went from 65 mph to 200mph while descending to land. I was cleared to land; and landed on Runway XX without issue. Corrective actions are as follows: clear the static port clog and inspect the engine with an A&P.