Assuming you have access to a gym with an assited pull-up machine I would consider using that to help grow into your pull-ups as well. Of course a chair could and would work too, but I just love using the assitted machine to help me verses a chair.

I would like to build up to a higher number of pull-ups so here was my routine the other day:

-Choose a weight that will allow you to do 10-15 (you choose your ideal number of reps in that range)
-Rest 30-60 seconds
-Reduce the weight and do your max
-Rest 30-60 seconds
-Reduce the weight and do your max
-Rest 30-60 seconds
-Reduce the weight and do your max

So in all you have done 4 total sets and on each your reps may have gone down, but your weight has gone up. You're still getting some assistance (you determine how much), and focus on the best form you can, and you are building your strength, endurance, grip and developing good form.

*I don't believe one program is right for eveyone, but this has really been helping me. I noticed that both my grip and form were week, and so trying to doing things unassitted was just making my form sloppy and my lack of grip strength wasn't helping me much either. This allows me more time on the bars, improving my forarm and grip strength, has helped get my joints and all of their connective tissues used to the movement and work load, helps me to focus on my form and not wiggle or kip all over the place, and I am beginning now to be able to do many more when not at the machine and they look better.

And yes, even losing 10-15 lbs has been helpful. Just less to pull :-)

All my best to you and your goals. There are quite a few videos on the web showing how most people that do not have access to an assist machine will start with just a little bit of help from a chair, and also from just starting at the top and releasing as slow as possible (working the negative). It's at least a place to start.

Again, just hoping the best for you and wishing you luck on your goals. Go get 'em :happy:

Back/Bicep, Chest/Tricep, Legs calisthenic split routines twice a week.

whyyyy? You will literally make no progress at all.
Splits are bad for beginners at lifting and even worse for calisthenics. Go read 'overcoming gravity' and get a full body workout 3x/week.

Also this might work on upping your pull up max:

I don't think it's that important when you do your training, except ofcourse if you really see less results yourself.
I have been training fasted while on a cut and I have been training in the late evening with as only food during the day being 4 eggs, some bacon and some lettuce that I ate. I never felt like I was too tired or that I didn't have enough energy.

The problem with splits is that you target your muscle 'groups' only once/week. VOLUME is more important than intensity if you want to gain strength or mass (for example: prison inmates and female/male gymnasts). Calisthenics is a lot of skill work and when you only train pull ups once/week the body won't care, while if you do them 3 times/week you will get stronger because you are practicing that compound movement and the nervous system is adapting so you can do it more easily.

http://i.imgur.com/nEFpi8D.jpg

For lifting, people take 5 min rests between their sets and for calisthenics this is the same, this is about the time that your muscles need to restore their ATP storage. If you do a full body workout, you might think you will be spending two hours on each training. Actually one hour is more than enough, because you can combine two exercises working on opposing muscle groups so you only take 2.5 minutes between sets.

http://i.imgur.com/2Xk6g6B.jpg

I hope these help, as you can see I kinda swear by this book. Steven Low is a bit the calisthenics version of Mark Rippetoe as 'Overcoming gravity' is comparable to 'Starting Strength'.

nope, that shouldn't make it a split that magically absorbs all your gains. You might even be able to get more out of the training because you have rested longer.