Grip the rings using a false grip, with the base of your palms on top of the rings.
Begin with a movement swinging your legs backward slightly.
Counter that movement by swinging your legs forward and up, jerking your chin and chest back, pulling yourself up with both arms as you do so. As you reach the top position of the pull-up, pull the rings to your armpits as you roll your shoulders forward, allowing your elbows to move straight back behind you. This puts you into the proper position to continue into the dip portion of the movement.
Maintaining control and stability, extend through the elbow to complete the motion.