Flat out I can do maybe 1 pull up. What can I do to improve my pull-up rep count other than "doing more pull-ups"? What muscle groups are used for pull-ups?
Since you can do only a single pull-up, I would suggest more reps with some sort of pull-up progression. Some examples are: jumping pull-ups pullups with your feet on the ground in front of you, leaning backwards assisted pullups (from a machine at the gym, with a rubber band, or by tying a rope to a weight that goes over the bar) Finally, you might look into kipping pull-ups, which require a little bit of skill, but allow you to do more repetitions, and therefore improve your dead hang pull up as well.
You can use the same sort of rep scheme that you would use for any other lifting exercise.
Dead lifts and lat pull-downs may help as well, but the above will probably be more effective.
The lats are quite involved with pull-ups. Also the biceps and a number of muscles of the upper body.
I went from be able to do exactly 0 pull-ups to doing 11 in one set and ~33 throughout a 90 min training session (among other exercises) in about a year, and that includes some holiday breaks and injuries that slowed me down considerably. I got the sense from reading about it and my own experience that there is NO exercise better for developing the strength to do pull-ups...than pull-ups themselves! (That said, any arm/shoulder/back strengthening will probably help some, and you should be doing that anyway).
What I did was first to just strain there at the halfway point for a while. Then I would do sort of "mini pull-ups" where I just pulled myself up a few inches and then back down for as many as I could do. Eventually I was able to do one. Then kept with that, and then eventually 2 and so on... I think patience is very needed for this project.