I recommend the Munhwa-dong G&M Clinic.

This is where I went for my skin problems. The nurses at the information desk were friendly. The doctor was also helpful, as he was more than friendly; he just gave the facts, making it easy to understand. The Google rating is low, but I was fine with it. What I wanted to know was how to prescribe medication and what kind of treatment would be best for me. It was nice that he answered all my questions.