Start doing 100 pull-ups per week, divided into 2-3 sessions. Slowly advance this to 200 pull-ups per week.
I like doing Every Minute On the Minute pull-ups, for 20 minutes. A good starting session for you is to do 2 pull-ups EMOM for 20 minutes. Next session, try 3 EMOM x3 minutes, followed by 2 EMOM x17 minutes. Next session, try 3 EMOM x5 minutes, followed by 2 EMOM x15 minutes. In this manner, you slowly advance your total pull-up volume by 2-3 pull-ups each session. The first 10 sets should be extremely easy, especially the first few sessions. As you advance, the final 5 sets will be extremely difficult. I used this method to advance from a max of 13 pull-ups to 20 pull-ups in a few weeks. At my peak, I did 6 EMOM x5 minutes, plus 5 EMOM x15 minutes, for 115 total pull-ups in a 20 minute session.
It's a fast and easy way to get a lot of pull-up volume in a short period of time. It worked easily when I was 165 lbs. Now that I'm 195-200 lbs, I can use the same method to get boosted from a max of 15 back up to 20 in just a couple weeks time. Never really tried to take it past that.
Also, I found that doing weighted pull-ups did fuck all to increase my max number of pull-ups. And I've done 100 lbs for triples at 200 lbs body weight. Doing weighted pull-ups makes me good at weighted pull-ups. Doing lots of pull-ups makes me good at doing lots of pull-ups.