Pull ups are normally performed with your hands facing away from you, while chin ups are done with your hands facing towards you (and chin ups also emphasize the biceps more). Chin ups are generally considered to be slightly easier version.
The following progression can be done with both pull ups and chin ups, even if you do not have a pull up bar (How to do pull ups without a pull up bar).
Start at whichever point of the progression is relevant to you.
Unless otherwise indicated, you should be aiming to complete 6 sets of 6 repetitions of the given exercise within a workout.
The first set is a warm up set: go back two or three levels in the progression and perform that variation for 6 reps.
Perform each following set ‘to failure’ (e.g. 6,6,5,5,4).
In each new workout, you should aim to improve on the number of reps you performed previously by at least 1, e.g.: (6,6,5,5,4) → (6,6,6,5,4)
Once you manage 6 sets of 6 reps, you are ready to move on to the next variation in the progression.
If you find yourself not progressing any longer, take a few rest days, then resume training.
Finally, if you are still struggling to achieve your first pull up, make sure to read How to do one pull up.