The Doctor
Question #1: Who is the Doctor?
Technically, of course, there doesn't have to be a Doctor. There could just be a group of Companions looking for the Doctor.
So, here are some ideas for the Doctor in this game:
Romana
Doctor-Donna
Jenny (the Doctor's Daughter)
The 15th Doctor
I'm open to ideas for other options.
Question #2: Does anyone want to play the Doctor?
Or would it be better if the GM played the Doctor? It probably depends on which Doctor we go with.
If we go with Jenny or the 15th Doctor, there's no continuity to worry about, so characterization would be whatever you want it to be. And Romana (I, II, III, IV, V?) would be almost the same.
Doctor-Donna would be the hardest to do, imho.