Like the Animal
The bunny hop is essential for clearing obstacles on a fast track....or as a climbing starter...It is really the same move as for the backhop, but moving along at any speed. Quite a useful move in street trials where everything is flat and square, but totally useless in natural biketrials where there is rarely any run up distance.

Maximum pull on the arms during extension
Start at a low speed...Stop pedaling and flex to prepare for an extension.
1° Start lifting the front wheel while pushing on the rear wheel, then full explosive extension, in one jump impulsion on the pedals at the highest position. This is for take off. 2° As soon as the back wheel takes off, fold back, pulling on the handlebars upward while folding the knees to clear the way for the bike. That is only what brings the bike up with you, no clip pedals. 3° Once the obstacle is passed, you can lower the arms and knees and let drop the rear wheel again to touch ground. If the obstacle is long, then you can keep the same position until landing flat.
See the move in a video

How to lift the bike in a bunny hop or back hop?
Lifting the handlebars is the only trick, because your feet can't lift the bike. It looks like the feet are stuck to the pedals, but in fact they are just limiting the ascension of the bike. For the impulsion, think about jumping on your feet without a bike. You need to flex the knees, then full extension, and then folding back in the air to get over the obstacle. On the bike, same business, flexing before a full extension from the highest position on the pedals (when the bike is standing on the rear wheel). During the extension, the arms are kept low to keep the rear wheel pushing on the ground, but as soon as the extension is over, it is time to fold back the knees with a synchronised pull on the arms upwards to lift the whole bike. All the move takes less than a second.
video
One handed bunny hop

video
Same with a mod bike
The higher you lift the handlebars, and the faster you push them forwards afterwards, the higher you can get the back wheel off the ground. Of course you need some synchronisation, this is the difficult bit that needs hell of practice. You can practice with full brakes on (the safest way to understand the move). That leads you to the back hop. video
practicing over paper boxes

Why aren't clip pedals recommended for this?
Because Its way too dangerous to have your feet stuck when you need to jump off the bike at any time for a safe crash. You'd rather land on your feet than on your back, believe me.
It helps a lot to have lets say a paper box to practice over, and build up the heigth progressively. There are loads of useless variations on the bunny hop, like the "one footer" or "no footer", or 180°. The next step is to land your back wheel on an obstacle, and stay there. It 's a variation of the bunny hop where you do exactly the same moves, but with much less speed, so that you land on the obstacle instead of passing it....see that in the climbing zone.