Pullup Progression

Focus on pulling your shoulder blades together.
Door frames are a great place to do this.

Focus on retracting your shoulder blades.
This can also be done by tying a knot in a scarf/sheet/towel and anchoring it in a closed door.

You don't have to jump, just get yourself to the bar in whatever way is easiest (stool, chair, etc.) and control your descent as much as possible.
Don't take more than 1-2 seconds to lower yourself (there are more difficult variations for that).

Lower yourself in a controlled manner to a 3 second count.
Use a stool or chair to get up on the bar if that's easier/more convenient.

Lower yourself in a controlled manner to a 5 second count.
Use a stool or chair to get up on the bar if that's easier/more convenient.

Chin should clear the bar at the top.
Shoulders and elbows should be fully extended at the bottom.

Use a grip that's MUCH wider than shoulder width.

Grip should be MUCH wider than shoulder width.
You'll still be using both arms, but focus on pulling yourself up to one side.

You should be hanging mostly from one arm with the other extended out as far as possible holding the bar with a two-finger grip.
Use the passive arm as little as possible.

You should be hanging mostly from one arm with the other extended out as far as possible holding a lower part of the towel.
Use the towel arm as little as possible.

Same as towel-assisted one-arm pullup, but let go of the towel and lower yourself on one arm to a 3 second count from the top.

Same as towel-assisted one-arm pullup, but let go of the towel and lower yourself on one arm to a 5 second count from the top.

Same as a pullup but with one arm.