Moon Golden Chest #1: During The Sword of Crota, follow the waypoints and keep an eye out for a large natural ramp. To the left of the ramp behind some rocks is the first golden chest.
Moon Golden Chest #2: During The World's Grave, at the bottom of the spiral staircase in the Hall of Wisdom will be some water. Follow the water to the second chest.
Moon Golden Chest #3: During The World's Grave, at the end of the mission, in the room during the final standoff.
Moon Golden Chest #4: Make your way to the Temple of Crota. After the large Ogre, make your way into the ship in the middle of the spiraling room. Look down in the middle of the pit to find the fourth golden chest.
Moon Golden Chest #5: At the end of Strike Mission "The Summoning Pits", behind the final boss Phogoth the Untamed, there's a landing pad with the final golden chest.