Run towards wall/rail with reasonable momentum. If you are going to speed vault with your right hand, then you should aim to take off with your right leg. If it’s your left hand, then take off with your left leg. I do it with my right hand, so in the picture, you will see a demonstration for a right-handed speed vault.

Take off with the appropriate leg, and throw your hand (which ever you are vaulting with) forward. Drive the trailing leg as well, as it is need to produce a small ‘kick’ towards the end of the move.

Your head is focusing forward on the landing spot now that you've finished the business of grabbing the rail. The trailing leg is driving over, whilst the speed from the run-up is assisting the body's progression over the rail.

Drive the trailing leg up to get height and speed and distance, kick the left leg.

Ideally - my body should be parallel with the obstacle. This is where the move gets tricky though; you have to pull your body back to upright, in order for a smooth, controlled landing. The aim is to land with the same leg you took off from...as you can see by my right leg coming forward. The other leg kicks backwards as you touch the obstacle. This enables the back leg to come forward for the landing. This is a very important part of the move, and if this is done well, the landing should be good.
Head is pushed forward to drag the body along in an attempt to get distance.

The leg you took of comes through and takes the landing. Try to make this into run in order to preserve ‘flow’. Then run off in a powerful manner, to make the move look as powerful, and speedy (hence the name) as possible. This landing shows my feet together, but you'll just have to trust the framing time! Depending on the distance, you'll find it harder to do a one-footed landing over distances.

Focus on the rail during your fast run-up. Leap with the same leg as the arm that you're planning on grabbing the rail with, and focus on getting the distance by pushing your upper body forward, and using the momentum from your trailing (but driving) leg to gain height. A decent landing makes a fluid move, and you can carry on to the next obstacle!