The "Dance King" is located in the Sunshine Club at the top of the Forest. When you try to go in, a robot will reject you for not having enough swag / style. All you have to do is go to the shop, buy the pair of sunglasses, put them on, and then you will be able to enter the club.
Once your in, just go over to the left and challenge him. For the actual challenge you just have to press the correct button at the right time. It's fast, and will likely take several tries, especially the last part. Be patient and try not to spam the buttons too fast. NOTE
- If you make any mistakes and continue on with the game, then you will have to start a new game because the robot will breakdown and you won't be able to challenge it again. I recommend pressing the Xbox button to bring up the guide and quitting out of the game. You'll be able to continue from the checkpoint right before the dance.
Mr Splash did a video for this one, but it doesn't explain how to get in the club. It just shows the actual gameplay. I'll provide it anyways in case people are wondering what that is like.