This is the only achievement in the whole series that requires you to play a new game+, but I completed it 50 minutes into that New Game+.
1) Before you hit the point of no return in your first game
, upgrade the weapon you would like to level up to at least V (5), my Avenger was at 6 I think somehow.
2) Complete your first playthrough of ME3.
3) Start a new game with that first character and carry over all your previous save's gun levels.
*Helpful: I turned the difficulty down to Narrative so I could run around and melee everything to work on Bruiser as well.
4) Run through the game to the point you get control of the Normandy. About 50 minutes.
5) After meeting with the Citadel Council for the first time, hit the Normandy's shuttle bay and level up a weapon to X.
6) BOOM.
It should cost 30-35K credits, which I made running through to that point in the game, not including what carried over in the new game+.