This Achievement is actually very easy to do, provided your group can comfortably get through Nightmare levels.
For starters, you CANNOT do this without four people. You can't tell the bots to sit at a chain and break them on command.
At the start of the section with the chains, as a GROUP, clear the Skaven guarding each chain anchor, the large mobs don't seem to spawn until the chains are broken, so clearing the few Stormvermin and few adds that are up the tower won't take long. After clearing the tower, send one person to each chain and attack them until they are ALMOST broken, once everyone has their chain nearly broken, give the signal and everyone will break them together. You have a 5 second window from when the first chain is broken to get this done. Once they're all down, move as quickly as possible to the big exit door that breaks open. If each person has a speed potion with them you will find this task much easier to do.
You don't have to clear any mobs, just run for the wagon at the end of the level and the Achievement will unlock at the results screen.