The only time Dolphus Raymond shows up in this novel is in chapter 16. The first time the kids see Mr. Raymond is on the day of the trial, and Jem wonders how someone can be drunk enough to fall out of his saddle so early in the morning.
We learn that Mr. Raymond doesn't socialize with the white folks in town, though that's more by their choice than his. He is seated with the Negroes who have gathered for the trial--clearly an unusual practice for a white man in Maycomb. Jem says:
"He likes 'em better'n he likes us, I reckon. Lives by himself way down near the county line. He's got a colored woman and all sorts of mixed chillun."
Dolphus is clearly an outcast in the town of Maycomb, and he understands why--he makes the fine people of the town nervous because he has crossed racials line and seems comfortable and content to have done so.
"'He doesn't look like trash,' said Dill.
'He's not, he owns all one side of the riverbank down there, and he's from a real old family to boot.'"
We discover he has a bit of a sordid past involving a former white fiancee and a black mistress and a shotgun (you can find that one yourself--what a story!), but "'he's real good to those chillun'...."
He is not, in fact, a drunk--his brown paper bag holds nothing stronger than a bottle of Coca Cola. When asked, he explains he does this drunk act in order to let others find something else to criticize him about other than the obvious racial issues.
In short, Dolphus Raymond is a character, literally, who emphasizes the obvious prejudice in Maycomb and demonstrates for Jem and Scout that loving people is far more important than loving the mindless traditions of hate.