To earn this achievement Henry must drink himself into a stupor. Stock up on as much alcohol as possible; spirits, wines, beers, meads, whatever you can find. At least thirty bottles of any type should be enough.
The next step is to drink as much of the alcohol as you can until you see the following notification:
Be sure not to consume any food as you are drinking. The more food you eat, the higher your nourishment will be, leaving less room for alcohol. Once you are stuffed, use the time dial to wait 24 hours. You can now consume more alcohol.
Repeat the process until you have drank enough to become addicted.
Note: If you have perks which make losing nourishment much slower, you can purchase or craft digestive potions to reduce overstuffing by 20, thereby allowing you to drink more.