Credits to UESP.net for additional info on this achievement. I also used "The ESO App" on android to track progress and for their awesome interactive maps.
Hero of Murkmire is awarded for completing five other large achievements in Murkmire. To earn this achievement, you must:
Complete 20 quests for Murkmire Adventurer: Quests can be found at all major cities. Look for the glowing quest markers to find quests that are not started. Story quests contribute to your total and are required for the achievement as well. Daily quests do not count toward the total. There are 26 quests available in the zone, so don't feel like you have to complete them all, as some (such as Monument of Change) can be time consuming.
Complete the Murkmire main quest for The River of Rebirth (This earns you The River of Rebirth achievement for Xbox as well). Main quest markers have a diamond with a smaller diamond inside and a "v" underneath.
Defeat both world bosses for Defender of Murkmire: There are two world bosses you need to defeat. The first, Baxilt-Gah is slightly south-west of the Blackrose Prison Wayshrine. The second, Walks-Like-Thunder, is to the north-east of Dead-Water.
Discover all Striking Locales for Murkmire Pathfinder: Striking Locales are marked on your map with an eye. If the icon is black, that means you have not visited the locale yet. If the icon is white, the locale has been visited.
Clear both Delves for Murkmire Cave Delver: The two Delves needed are Teeth of Sithis, which is east of Dead-Water, and Tsofeer Cavern, which is south of Dead-Water.