You could keep your bodyweight mass down, lose fat if you have any, and do several lat exercises such as landmine rows with t bar, or lat bars, one armed rows, etc. Use an endurance type rep scheme(15-20). Working your forearm strength and grip as well as core and biceps to some degree with additional training. Try adding weight here and there but focus more on increasing reps since this is your goal. Outside of this.. just do 6 sets of amap reps of pullups a couple times a week and try to increase your numbers each time. If you fail, stop and start a new set. At the end of the month., try the pull up challenge again. Eventually you'll hit a very high number of reps to where muscular endurance and cardio are going to be the factors that get you that high. You will have to truly work on yourself, and be very lean. Try to not gain too much muscle, even losing muscle in legs might help you.
Reevaluate where you are at the end of each month.. what breaks first, your grip, energy, back, shoulders? And work on those things first.
To reach record high nymbers, you'll have to lose as much weight as you can while keeping your muscles strong. You'll potentially even have to lose muscle in places that don't matter just to she'd a few pounds off such as chest, triceps, legs, etc..
Genetics can play a key role too.. for instance a 5'4 person is going to weigh less and be more mechanically advantages to do more pullups than a 6,0 person