The two best upper body calisthenics exercises for overall strength and muscle building are the pull-up and the dip. They’re best because of the range of motion allowed and the strength required to lift the weight of the entire body. Being vertical push and pull exercises (although some people do not consider the dip a vertical push because of the possibility of forward lean) the posture at the beginning of each movement allows a full stretch and at the end of the movement a full contraction with the weight of the entire body.
Both of these movements are required for a complete calisthenics training program. The problem is that many beginners will not be strong enough to perform the exercises with proper form and enough reps to stimulate strength and muscle gains.
The solution to this problem is to practice the Jackknife variation of each movement. The Jackknife variation preserves the range of motion and the direction of movement but takes some of the body-weight and stability requirement out of the equation. In the Jackknife pull-up (or chin-up) and the Jackknife dip, the feet remain in contact with the ground throughout the exercise. With the easiest variation, the feet are flat on the ground and the knees are bent, and for a more difficult variation, the heels are touching the ground and the legs are straight and you hinge at the hip as you are performing the movement. An added benefit is that you are able to use your legs to assist yourself if needed. To make each exercise more difficult, keep the legs straight and elevate the heels on a bucket or short table.
To perform these movements properly, you will likely need a set of gymnastics rings or other suspension trainers that can be adjusted. For the Jackknife dip, adjust to a height that is just below your waist and for the Jackknife pull-up, adjust them to chest height.