This is very simple.
Load up Act 1 Mission 2 at the rock bar. Interact with the dart board, move the case in front, get darts from Bartender and hit 5 bullseyes.
Now for the achievement you need to successfully complete all stages of the drinking game. There are 4 stages, if my memory is correct, and you must have "WON" appear on screen after each stage of the drinking game. If "LOSE" appears on your screen at any stage, you must restart the level and try again. Doing this successfully will unlock Professional Drunk.