Complete the quest "The Crumbled Chamberlain" within 10 minutes of accepting it.
I did not find this achievement particularly difficult on the PTR, but it may be far harder on actual Live realms. The pieces of the Chamberlain did not appear to have a great respawn rate, and their respawn positions are entirely random.
I recommend looking at the map on the The Crumbled Chamberlain for a list of all the locations that you need to reach within the 10 minute timer. Just run by all those points and if a gear doesn't appear on your minimap, keep going. If it does, stop and find a way up quickly.
Remember that you have to end at the cave under the Court of Bones, so I recommend starting at the troll ruins first and then going clockwise to the Court of Bones. It had approximately 2 minutes 40 seconds left on your first try; the in-game achievement tracker does track the time for this successfully.
Kind of implied by Simca's comment, but to state it explicitly: the quest objects are shared. This means that if another player beats you to the object, then you must returning to searching its spawn points. "Quests: Crumbled Chamberlain: Statue pieces now do not despawn when looted, allowing other players to loot the statue piece as well." (source)
It is definitely useful to have an eye on the map WoWDB provides for the quest (link in Simca's comment). Tip: All four quest objects are in places where you have to figure out how to get at them. If you don't see it, try looking up first. Still don't see it, then try looking below on a ledge or something.
Abandoning the quest will not reset the achievement timer. (Quest timer ≠ achievement timer.) You will have to wait for the timer to reach zero before making another attempt.
Note that this appears to be a weekly quest.
It is indeed a weekly quest.
Seeing how the pieces are placed, this quest can't be accessed until your realm completes Stage 2.