Instructions
Walk away from the Blackthorn Gym master, Clair, to the moving platform that brought you to her. Step on the arrow pointing back the way you originally came to move the platform toward the gym entrance, then exit the gym.
Walk back behind the gym to the body of water nearby. If you don't have any Pokémon in your party with the surf ability, go to a Pokémon Center and take one from your bank. Of the six spots in your party, leave one empty so that you can receive a Dratini once you enter the cave.
Move your character directly against the shoreline, access your pause menu and use your Pokémon's surf ability. This places you in the water on the opposite side of the shoreline.
Surf directly north to the entrance of the Dragon's Den and enter. Though there are trainers in this cave you can fight, it isn't necessary. Go to the back of the cave and speak with the elder to receive a Dratini. Clair appears from off-screen afterward to award the Rising Badge and the special move "Dragon Pulse."