Baby steps. Beating other bots in deathmatch is a good start. I love that they only use the rocket launcher, giving the careless bot an equal chance of blowing itself up.
Parsing what's on the screen in Doom is potentially a lot easier than in modern games: since there is no texture filtering or anti-aliasing, and due to the 2.5D perspective, most vertical runs of pixels on the screen map exactly to (linearly?) scaled columns in wall textures or sprites. I would not be surprised if you could come up with a fairly simple algorithm to determine the exact position and orientation of the player and the objects on screen within the map, without any real AI/learning involved.