You can tell hes making up the story on the spot by the way he looks up to his left when thinking. When people do this, they tend to be thinking of a lie. 0:41.
Never really realized until now that from 0:42 - 0:46 Joker was thinking about a made up story about his father being a drinker and a fiend in order to falsify his reasoning behind the scars. Damn, what an unbelievably incredible performance by Ledger.
You can tell hes making up the story on the spot by the way he looks up to his left when thinking. When people do this, they tend to be thinking of a lie. 0:41.