Atticus later explains to Jem the differences between his actions with the mad dog and Mrs. Dubose's fight against morphine addiction (in Chapters 10-11 of To Kill a Mockingbird).
"I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. It's when you know you're licked before you begin but you begin anyway and you see it through no matter what. You rarely win, but sometimes you do."
The two chapters illustrate two supposed acts of courage: Atticus killing Tim Johnson, the mad dog; and Mrs. Dubose's struggle to kick morphine. Atticus discounts his own skills with the rifle in spite of his children's adulation and discovery of this trait (and his old nickname, "One-Shot Finch"). It is something of which he is not proud. Mrs. Dubose's actions, however, were deliberate and with the prior knowledge that it would be a hard and difficult struggle. To Atticus, "She was the bravest person I ever knew."
I agree that these two incidents are supposed to show Atticus' character. In each instance, the quality that is most prevalent is Atticus' quiet sense of courage.
He faces the dog and is completely calm. He steps up without the confidence of his children and accomplishes exactly what he plans to do.
Both the children are afraid of Mrs. Dubose. Atticus however, is not. He treats her with the same calm yet deliberate attitude that he faced the mad dog. He knows exactly what to do and is confident in the outcome.
For both the children, these are moments of witnessing a side of their father that they 1) do not understand, 2) do not wholly trust at first and 3) allow to boost their confidence and respect in him.
In my opinion, the connection between the two is that both of them show the kids learning about what Atticus values.
The case of the dog shows that Atticus does not think that a man needs to act all tough. He turns out to be a great shot (which Jem thinks is very manly) but he does not show it off and has not used a gun in years.
The case of Mrs. Dubose shows that Atticus thinks that it is more important to "turn the other cheek" than to get revenge. He values the idea of helping those who hurt you.
These two incidents both show how Atticus thinks people should live their lives.