Instructions
Check the town's bulletin board every day. Upcoming store specials are listed on the board. Wait for the Treasure Chest to be announced as an upcoming special.
Go to Tom Nook's store on the indicated special day. Pay Tom Nook 8,600 bells for the Treasure Chest.
Walk around the town, and look for Wendell the walrus if you do not want to wait for the Treasure Chest to be sold. Wendell can sometimes be found wandering around on weekdays, between six in the morning and noon.
Speak to Wendell, and he announces that he wants food. Go down to the shore, and fish for a King Salmon or a Ray.
Return to Wendell with your catch, and give it to him. In exchange, he rewards you with a free Treasure Chest.