The handstand push up is definitely an advanced movement. Before attempting, please be sure that you have built up the prerequisite strength to hold a handstand against the wall.
To perform a handstand push up, you will need some free space on the wall. Some people like to place a mat on the floor to cushion the impact of the head touching the ground. An AbMat or gymnastics mat can be useful here.
Standing facing the wall, place your hands 6-12 inches in front of the wall. The hands should be slightly outside of shoulder-width apart. Fingers can be slightly turned out.
From here, kick up into your handstand making sure that your heels come into contact with the wall. Once you're inverted, squeeze your core, quads, and glutes to find a rigid position.
Lower your head towards the ground with control until your head makes contact with the floor or mat. Avoid the elbows flaring out to the sides. Generally speaking, a 45 degree angle with the elbows is an ideal position.
Once you've made contact at the bottom of the position, push back up maintaining that rigidity through the midline. The rep is complete when the arms are fully extended.
This is an advanced movement. To build up strength to perform this exercise, get comfortable with handstand holds.
You can also perform "negatives." This is where you would drill the "lowering" phase of the handstand push up - starting from the top of the rep and lowering yourself very slowly towards the bottom. Come out of the handstand at the bottom.
This is one rep. Perform 3 sets of 5 negatives at a slow tempo to help build up the push up strength.