Mrs. Dubose appears very annoying and opinionated. She also appears cranky. According to Scout, she is the "meanest old woman who ever lived." She yells insults about Atticus to the children as they pass by her house. She is a bully towards them, and towards Atticus.
Atticus is tolerant of her, and actually respects her. Her "mean" attitude is a result of a morphine addiction, brought on by a terminal illness. It isn't the addiction itself, but the withdrawl - she is determined to kick the addiction before the disease kicks her, so that she can die free. Atticus believes that this is true courage, and he supports her.
One of the ways he does that is to have Jem go and read to her. This is punishment for Jem's attack on her garden, but it is also a gift to her - the reading is a distraction from the pain she is in, and allows her to go further each day without getting any morphine. In essence, it is what helps her to actually quit the addiction. Atticus deals with his kids by showing them that they need to understand what other people are going through and to support all people. He deals with her by being kind and showing respect for her fine character, no matter what she says about him.