This isn't at all difficult, even though you have limited time. I wouldn't recommend doing this your first time through, as you won't have all of your weapons. By playing further into the game, you'll be able to get all of your weapons, your upgrades and you'll be familiar with what weapons/combos you like best.
As long as you don't fall off and don't miss any enemies (make sure you kill them before moving on), you should do this with time to spare. If done correctly, the achievement will unlock once you've used devil pull on the end to enter the door.
If you're planning on playing the game through on Heaven and Hell difficulty, this can be done with almost 0 effort, as all hits are a one hit kill.