I've watched videos on YouTube and have read/watched the solutions on here. It seems that nobody really knows when he appears but I will say from my personal experience, I unlocked this on round 9. This might be coincidental but I will go into details with my experience. I tried this on solo play and was unsuccessful as I never spotted the doctor and I went as far as close to round 20 so I decided to try on Xbox Live Custom Games (I won't confirm if this only works on Xbox Live because I'm not totally sure but since it's an easter egg, I decided to try). I never spotted him on any of the rounds between round 1-8 but at the very beginning of round 9, the doctor spoke so I had a feeling that I was going to find him. I saved 3 zombies on the round and found the doctor in the morgue. He was behind the window in the operating room. I shot him with the Type 100 while being fully loaded and the achievement unlocked while I was reloading. Now I'm not sure if him talking in the beginning of the round was an indication that he was going to be doing his rounds, but like I said, that's the round that I was able to find him. I didn't listen to his message because my volume was low. Enclosed in the video below is all the possible locations where you can find him. It's not my video by the way but it helped me to find him whereas other videos didn't show as many locations and this video gave me the idea to check the operating room.