Q: How does a person work out whether they're an alcoholic?

The simple answer is contained on page 44 of the book Alcoholics Anonymous:

"If, when you honestly want to, you find you cannot quit entirely, or if when drinking, you have little control over the amount you take, you are probably alcoholic. If that be the case, you may be suffering from an illness which only a spiritual experience will conquer."

Going to AA meetings and listening, talking to AA members, reading the book Alcoholics Anonymous, and working the First Step of the Twelve Step Programme will help answer that question fully.