Ladders. Do a pull up come off the bar, do two pull-ups, come off the bar etc. Can work your way back down the ladder or not depending on your strength levels, time, and what training routine you’re on right now. Start by doing this twice a day and increase to whatever is reasonably possible for you in a day.
I would advise using different hand positioning as well. So do one ladder one way and then another way for the next ladder. I assume you’re on some type of weightlifting routine so on your typical “back” day or however you have it split up. I would say at the most only do one vertical pulling exercise or use that day to go really heavy on the pull-ups. Other than that do alot of horizontal rowing.
One thing that seemed to work well for me and alot of guys on my boat getting ready for PT tests is if you have access to a kettlebell or could use a dumbbell but most people won’t have a single dumbbell lying around in their bedroom or something. Maybe I’m weird for having a kettlebell in my room, but use the kettlebell to do a clean and press then a pull-up using the ladder method.
Using ladders like this we didn’t generally go to failure on our top set. So if we knew we were good for a top set of 10 we’d normally only go to 8. Then 3-4 days later increase it to 9. Once a week or so you can do a kind of test and max out as well. But the idea is to get in as much frequency as possible.
Since you’re going for weighted pull-ups as well you’re going to have factor that in for your progression as well so whatever number of reps you’re trying to get to with a given weight then cut down the reps and add weight.
The advice given is good if you can’t really do pull-ups but since you’re trying to do weighted pull-ups I assume you can already do more than 10, so make sure your form is good but you don’t want to sit there and squeeze for 1-2 seconds on every rep. You want to control your reps sure but that will burn you out fast if you’re shooting for maximum reps.
Obviously this isn’t the only way to do it but I’ve seen it work with alot of people, hope it helps.