While fairly straightforward, there are a few things to keep in mind when going for this achievement. The earliest you can get this is once you reach player rank 3 (assuming you aren't in a party with a higher ranked player who buys the drinks for you). That is when you unlock the first two drinks at the Abyss Bar on the Spacerig. You'll want to keep buying Oily Oaf Brews until your character starts to wobble on their own and parts of the screen get blurry. Then, just make your way to the barrel hoop game to the right of the bar and jump through the hoop.
How many beers it takes to get drunk depends on the potency. The Oily Oaf is "safe for work" meaning it will take multiple beers to get drunk (about 4, so you'll need at least 150 credits or so). The Leaf Lover is "buzzkilling" and will instantly make you sober again so make sure you don't buy it even though it is cheaper.