Instructions
Travel to the Goron city by following the Death Mountain Trail from Kakariko Village. The Goron city is located in the northeast corner of the Hyrule map in a cave on top of Death Mountain.
Go down the steps to the bottom level of the city. Stand on the mat in front of the sealed door and play "Zelda's Lullaby" on your ocarina to open the door.
Go through the door. On the other side, talk to King Darunia. He requests a song. After the dialogue, play "Saria's Song" on your ocarina. After the cut scene, Darunia gives you the Goron bracelet.