I do them.
The best way is to flip your legs up against a wall into a handstand position. Once up, move around until you feel balanced, stable, then slowly start to lower yourself down low as you can, ideally until your head touches the ground, then press yourself up to the starting position.
A good first goal is to be able to do 5 good ones, head touching the ground. At first you might have to lean your legs back, to rest on the wall, to help maintain your balance, but after awhile you will be able to keep your torso and legs straight, like a reverse standing overhead press. When you can do 15- 20, try them on two sturdy boxes or small balance balls (the blue ones that fit in your hand) or when you get really good at them you will be knocking them back from a freestanding position. In my opinion handstand pushups are the king of pushups, because they work your entire upper body: all three heads of your shoulder, your triceps, traps, biceps, some chest. They also have the added benefit of dramatically improving your posture (when done perfectly straight) because they force you to stabilize your upper and lower body, using a lot of lower back and ab strength.
But a word of caution. Warm up well before doing these, especially your elbows and shoulders. Also, the more flexibility you have in your wrists the easier it will be for you to balance yourself verticaly.