Just caught your post on the news feed, thought I'd drop in. Pull-ups are a really good bodyweight exercise, in my opinion the best one for upper-body strength.
A few tips for increasing the number you can do:
i) Get someone to assist you, by pushing up underneath your legs, continue this over the days (i.e. get a training partner), until you can do a few more, repeat.
ii) Do negative pull-ups, what this entails is holding yourself up at the bar (jump if necessary), then slowly lower yourself down, perform reps of these in the same way you'd do pull-ups (i.e. pyramids). These are a stinger, but they'll help.
iii) Do multiple sets of lower reps, (i.e. 10 reps of 1 pull-up), anything to get you doing more, the higher sets will come later on.
Speak to your Unit PTI, that's what they are there for. Ask for a training programme and if they is worth their weight as a PTI then they will provide you with a schedule that focuses on the key elements involved in passing your PFA.
But seriously, how old are you? It shocks me when I see lads of under 21 struggle with 1.5 mile run in 10.30.
I have just had an oxygen thief wanting to join my unit, ran it in little under 20 minutes and then couldn't understand why we told him he needs to sort his fitness out and come back in 6 months.
Easy fellas, my phys was shite when I first joined, the guy just needs a bit of encouragement.
I'd recommend you mix your training up mate, get some hill sprints in, longer runs (3 miles plus) and try setting yourself some circuit training up. Also perhaps try taking up a sport if you haven't already, muay thai boxing is particularly good as it is an all body workout.
Reference pull ups, you'll only get better the more you do, there are no shortcuts. Put in the effort and reap the rewards!