Very easy to obtain!
Just follow the Youtube video; with the fire alarm going off you can basically just run through the whole level with the fire axe equipped if you're in the fireman costume. Use the fire axe to kill the main target instead of the fiber wire.
Achievement pop and Silent Assassin rating to go with it afterwards if you time it right.
Enjoy!
This is not my video, full credit goes to the guy/gal behind the 'HitmanVideoGames' channel on the tube.