Handstand Press
Below is the technique for a simple handstand press. This means that both your arms and legs will be bending to make things easier. I recommend having the handstand fairly solid before you try this skill, as well as a couple handstand pushups against a wall.
On to the press.

You'll want to get into a tuck position and put your hands out about shoulder
width in front of you. Take note that your shoulders are higher than your hips
right now.
Now what you're trying to do next is to get your hips higher than your
shoulders. We'll do this by bending the arms. There will be tension in your arms
and shoulders at this point, as you are now holding your bodyweight in the air.
You'll also start to feel your fingers balancing you in the position, just like
a regular handstand. Keep yourself tucked. And despite what it may look like (?) don't rest your
knees on your elbows. Your elbows are close by, but provide no support.

In addition, don't bend your arms too much. If you bend them to 90 degrees,
you'll mash your face, as you can see below.

But trust me, if you ever do this you won't the next time. It's a fast learning
curve.
When you've bent your arms to the correct angle (a bit greater than 90 degrees),
and your hips are higher than your shoulders, it's time to press up and begin to
straighten out your body.

The most important thing I can say here is KEEP YOURSELF TUCKED!
If you open your legs before you reach the position shown above, it's called "planching"
the handstand. This means that you're extending into more of a planche position
then a handstand position. This is obviously harder, and will probably cause you
to fall down and fail the press. If you feel like your legs are always weighing
you down in this skill, pay attention to where your hips are, and where in the
press you are opening up the tuck. You want to keep yourself tucked until right
after this position, then straighten out your arms and legs....
Bam. You're in a
handstand.

What to do if something goes wrong
While learning this, even though your handstand may be good, and you're strong
enough to press, you'll find linking the two together takes some work. So what
do you do when you find yourself overpressing and falling to your back?
I hope you know these safety measures from regular handstand training, but if
you don't, here they are again. Use them everytime you start to fall towards your
back. They are safer than merely slamming yourself back-flat into the ground, or
trying to put your feet down and slamming into a bridge (ouch!)
First, you can roll yourself out of trouble. When you feel yourself start to
fall, TUCK YOUR CHIN TO YOUR CHEST, and then ROLL OVER YOUR SHOULDERS, NOT YOUR
HEAD. It's just a simple forward roll.

Umm... I'd also recommend trying this where you can roll away safely. The space
you see me in is a bit cramped.
The second option, and the absolute best option I recommend if you're working on
parallettes, is to pirouette out of trouble.

What this entails is turning yourself 90 degrees to one side (the right side in
this case) while you push off with the far hand (my left hand in this case).
With any luck, you'll land on your feet.
You can see why this technique is ideally suited for the parallettes. If you
were to try and roll, you'd end up falling in-between the bars. By turning
yourself like this, you can effectively turn yourself out and away from the
bars. This will save you some pain.
Of course you can use this on the ground, I actually prefer it to the roll, but
know that it's the only technique you'll use if you take things to the
parallettes.