First, we will establish that he actually doesn't drink. The bottle in the brown paper bag that he has is a bottle of Coca-Cola. However, he uses the brown paperbag to maintain for the town the image that he is a drunk.
Dolphus goes head first against society in his personal life choices. In a segregated and racist society, he has chosen to live with and have children with a black woman. He is like Atticus in the sense that he is deliberately challenging social standards.
However, he doesn't have Atticus' courage. He can't stand in front of the mob like Atticus did in front of the court house and fight the town belief structure. He pretends to drink so that others will avoid him. Believing him to be a drunk, they assume that his "poor choice" in his relationship is a result of his drunkenness. The town leaves him alone. This is how he wants it. He is trying to avoid what he himself calls,
"the simple hell people give other people—without even thinking.”
You could say he is a coward, or you could just say that he is trapped in a system that won't let a man make his own choices.