The body is quite upright, but as you get better, you find yourself leaning back as it makes the move look more 'efficient' by getting your body close to the rail.

Once again, leaning back is a personal preference, but it does help if you're having trouble. The main part of the move is the drive up of your [right] leg (the one closest to the rail). Your hand is on the rail, guiding you, but not providing a massive amount of force.

Right leg is driving up, left leg gives a little push to provide the sideways momentum (although this is mostly provided by the driving leg and guiding arm).

Arm starts moving backwards, but there's no rush, this is lazy...:P.

This is where leaning back makes a difference or not. For simplicity's sake, I'm quite upright - but you can get a lot lower. Don't worry about missing the rail with the hand that goes back - because you're not going too, and if you do, it doesn't matter; you'll be over the rail by the time your hand misses.
However, it's very hard to miss the rail, because your body subconsciously keeps a track of where it is - your other hand is holding it, and so you instinctively know where to put the other - if you're still doubtful, just do it without thought and your body will do the rest.

I tend to keep my legs up quite high, and give it a little kick out towards the end to 'pull' myself away from the rail for some distance. It's up to you what you want to do in the air...from tucking to a rocket-vault shape, they're pretty much all possible.

Your arms don't do much except hold the rail, and keep you going straight. Once you've caught the rail with your second hand, the other can let go.

Single footed landing for a good run-off, and you're done!

This vault has a magnificent feel to it, you kick up with the closest leg to the obstacle, and transfer your weight across your shoulders as your other hand finds the rail. It can be incredibly fluid as well - as well as individual...the positioning of the body in the air means a lot of adaptations can be made (including a turnover/turn vault).