Case 1 Achievement:
On the island you will have to collect all the explosive ingredients for the lobster, once you have them all you can access a cave and inside you will speak to finley. After this scene you will head out and start the dance competition, simply wait for it to finish and it will give you a selection to choose your winner for this achievement.
If you need more help with this you can follow it along in my guide below which will start before collecting the ingredients: