Buy one of the keychain video recorders (2 buttons, rechargeable battery). Take it apart, mine had 4 screws (2 outside, 2 inside).
Clip the battery wires.
Glue an L shaped piece of plastic or a coated wire to the board,and to the back of the camera to stabilize the camera.
Open your helicopter up, and clip its battery wires.
Carefully spice (twist and tape, or solder and tape) the wires, 3 reds together, 3 blacks together to form a parallel circuit.
Put the helicopter back together.
Sling the camera board under the helicopter using a rubber band.