If you are looking at a fantasy/cod-medieval setting, you don't have much by way of a social security system and very few people have disposable income. So unless your character is the rakish son/daughter of the local lord (like an alcoholic version of Prince Harry in Henry IV), he or she would need to work to drink. Probably unskilled manual work or agricultural labouring if the drink is taking hold. If they were a tramp or a pauper, they'd have no income (unless you have an enlightened poor relief system) and would be begging, stealing and generally making a nuisance of themselves to get drink.
Interestingly enough, the character in question is (or was) the ruler of the kingdom in which the story takes place. I should probably give a little backstory:
Our hero(ine) loses her father (the king) around the age of 10 and was to ascend the throne as the next ruler, but a corrupt court of advisors pretty much said "no way." She was replaced by one of their own children so they could divide the kingdom up amongst themselves without anyone to challenge them. They offer someone a generous payment to take her away and "get rid of her," but something happens and she ends up being sold off as a slave instead of being outright killed.
So for the next 6-8 years, she spends her time as a servant to rather unpleasant people, and it wears away at her until she gets thrown out for being less and less productive, ending up a hopeless (and homeless) alcoholic.
Until the supporting cast meets up with her, I need to figure out how to support her habit, so to speak. Oh and yes, the setting is pretty close to the typical, medieval fantasy fare.
Hidden still (moonshine), of course, begging, bartering - does she have any special skill she'd be willing to exploit for alcohol?
Any special skills? Hmm, skills of note? As nobility, she can read/write (unlike most commoners) but in towns/cities where nobody else can read, that does little good. The royal family (upon reaching adulthood) receives a boon that makes them highly skilled in martial arts without any formal training. Those are the only two I can think of that would be relevant to commoner life.
Sorry, I had to kinda rush through this, gotta get ready for a final.