Cause the kite to hover in front of you for an instant by using a snap stall (rock the kite from side to side by pulling left then right, to get the feel of it). Once the kite is stalled, push your right hand slightly forward and immediately pull your right hand far back and extend your left hand well to your front. This will cause the kite to flip. After you have that part down, combine all of the parts together to for one flowing motion. Some kites need a gentle motion of the pulling hand, others need a snappy pull...experiment ! Here's a video clip of the axel performed by Dodd Gross. ("avi" format (2.2 Mb) you can also get it here (U.K.) if that's closer to your habitat).
Another approach to learning the axel is to fly the kite to the edge (say right edge) then initiate a slow push turn by pushing the lower (right in this case) hand gently. As the nose passes 12 o'clock snap the right hand back and extend your left hand forward. Once you've mastered this, try the approach above...

What seems to be essential for learning a nice axel is the setup as present in both ways described above. You need to have the wing you are going to pop to be slightly back from the other wing. Look at it this way: the more the wing you pull is towards you, the more you will pull wing AND spine, and the kite will start moving forward as well as rotate. That's not what you want. Having the pop-wing leaning too far back results in something similar. A slight leaning back is just what you need.