if you're doing weighted pullups you're probably on the right track, however there are a lot of factors that apply.
1. first, as many have stated, rest is crucial. unlike running or working your abs, your lagre muscle groups need rest. work out your back, biceps, and forarms all on the same day, every other day at the max. (only recommended if your pft is within a month) otherwise rest for two days b/w back workouts.
2. plateauing is so common. it's happened to all of us, i don't care who you are. the reason why our muscles plateau is they become used to the exercise. you can counter this by using variation.
here's a suggestion: alternate b/w the following workouts A and B
let's say for back workout A, you do weighted pullups. you can start by using a lighter weight, such as a 25 lb plate on the first set and work your way up to a 45 lb plate for your last set. do each of these sets to failure plus have a spotter help you with 2 or 3 ASSISTED REPS. (this is where you're really gonna get stronger) the assisted reps should be harder than the normal ones. make sure the spotter is not helping so much that he makes them easier. this will be very painful, but tough it out and you'll see results. the next time you do workout A, you can start with the 45 and go down to the 25 for a little variation.
now for workout B, do 2 sets of unweighted pullups at 10 reps each. then for your last set go to failure plus 2 or 3 assisted. the combination of variation and forced reps will shock the muscles forcing them to work harder and get stronger.
also, when you work out your biceps and forarms, make sure to do burnout sets. this will condition your forarms, which are the first to give out while doing pullups.
as for alternating b/w palm-out and palm-in pullups, i would not recommend that. here's why: the palm-in style works the lower lats, while the palm-out works the upper lats. so doing the palm-out way probably won't make your lower back any stronger, or for that matter make palm-in pullups feel easier.
3. last thing - psychological factor: 20 is a large number when thinking about pullups. don't get burned out by this. maybe set a goal to do 12 weighted pullups with a 45 lb plate. 12 doesn't seem as bad as 20, however if you can do 12 pullups with that much weight, 20 with only your body weight should be a dream. also, when you're doing your pullups and it starts to get hard close to the end, don't let yourself give up. the body naturally wants to quit, but you have to fight through it. i like to meditate on my arms, imagining they are hydrolic lifts doing all the work. it sounds silly, but it works for me. i know sometimes i've been so pumped up that i squeezed out about 3 more reps when before i would give up and get only 1 more if i was lucky.
these are some things to think about
good luck!