Although I'm not an expert, I've had a lot of experience weight training and doing calisthenics. The first thing to note is that gaining muscle is not necessarily synonymous with the amount of pull-ups and push-ups you do. This is a common belief but there are many skinny people who can do a lot of push-ups and other body weight exercises. This might be a long post but it will hopefully help you.
I was in the same situation as you so let me share my advice. I started out about 5'10 and 120 lbs. I wanted to gain muscle but had no gym access as well so I bought a pull-up bar. I worked out about 3-4 days a week doing different push-up and pull-up routines. I eventually weighed in at 130 lbs but went from 30 max push-ups to 70 and 28 max pull-ups from 5 in the course of about 7 months. As you can see, I packed on a small amount of weight but was able to do a large amount of repetitions. This is because of the amount I ate and the exercises that I did (I'll come back to this). I then got into weight lifting (trying to gain muscle) and ignored calisthenics for the most part for several months. I then weighed in at 153 lbs. (hovering around 10% bf the whole time) and had a 23 max pull-ups and 50 max push-ups. I gained muscle but was able to do less repetitions. Why? It was all about my training and diet.
I don't know how much you know but I'll try and be as elementary as possible. You don't gain muscle by simply working about, it's mostly DIET. If you eat more calories than you expend, you have a caloric surplus and gain weight. If you eat less, you have a caloric deficit and lose weight. Muscle is complicated, but the amount of it you have is not completely relevant to strength or endurance. I'll come back to diet but I need to discuss training routines first.
Simply doing push-ups and pull-ups doesn't make you gain much muscle as you can see from my experience, it's a complicated system with many factors but muscle is gained by tension and diet. This is why body weight exercises don't allow you to gain muscle very fast, over time your body becomes accustomed to the weight and it only builds endurance. Weight lifting allows you to increase your tension in the muscle by adding weight to constantly challenge your status quo. Weight lifting can help increase your rep count if you do endurance exercises, but weight lifting for strength or hypertrophy (muscle enlargement training) will help you increase muscle more but not rep count.
Before I get into your routine, I'll talk about diet. If you decide to weight lift and/or do calisthenics, you need to count your macros. Carbohydrates are your main energy source and are found in wheat products and grains. Protein plays a role in muscle growth. Vitamins help kick start your progress, my weight training teacher explained it in terms of cars. A sports car can't run well without premium gas, vitamins are basically the premium gas in your body if that makes sense and it allows you to perform at your full potential. Fats are important for health in general except avoid trans/saturated fats. Make sure you get enough of each to perform well. If you don't get enough carbs, your body will use protein as an energy source and your muscle growth stagnate.
Here's what I recommend for your routine, I know you want to build muscle but you also need to increase your rep count for the CFA. Try weightlifting to gain important overall conditioning and also gain muscle. Do calisthenics in your routine to increase rep count and NEVER quit the basics, ignoring these will make your rep count worse. If you don't have gym access, try investing in a home pull-up bar and dumbbells. I would really try to get gym access though. If you do get into weightlifting, try to avoid one type of training. Training for strength can be useful but don't help you with the endurance needed for military training.
Here's a sample routine of the program I do:
1.
Chin-ups (total reps) =
Chin-up max + 50%
2a. Front Lat Pulldowns (Wide, overhand):
2 x Max Reps(20-25 Reps 1st Set, 10-20 2nd Set)
2b. Standing DB Military Press:
2 x Max Reps (15-20 Reps 1st Set, 10-15 2nd Set)
3a.
“Rolling” triceps extensions
– 4x8
3b. Hammer Curls
(both arms same time)
– 4x8
4. Core Circuit: x2
A) Toe Touches x20 B) Bicycle x 30 Total C) Side Plank x 30sec each side 5.100 push-ups as fast as possible:
Anyways, this post is getting long so I'll just link a beginners guide. Also try checking out "barstarzz" or "calisthenics unity" on Youtube, I used their routines and saw lots if improvement in my rep count.
http://liamrosen.com/fitness.html#part4
Let me know if you have any questions, I can't answer it all in this post.