Jerry Cruncher is a grave robber. He digs up corpses from graves and sells them to medical students so that the students can practice on them. He does figure into the plot of the story by going to Paris with Lucy and Jarvis to get Dr. Manette and he helps Sydney Carton in London obtain information on Barsad, the spy. His character provides some comic relief to the story, in a twisted sort of way. He justifies his gravedigging work as noble, but regards religion as weakness. At the end of the novel, he decides to give up grave robbing after being affected in a positive way by Darnay and Carton. This makes his wife extremely happy.
You can read about it here on eNotes.
There was a long-practiced method of getting bodies for the scientists and doctors to practice on, and it didn't involve getting the consent of the deceased prior to them kicking the bucket. Jerry Cruncher goes out at night and digs up bodies so that he can sell them to doctors, professors of medicine, etc. for use in classes, examinations, studies, etc.
The interesting part of this is that Jerry considers what he does to be an absolutely productive and important part of society and of life in general while he regards prayer and other such religious activities as a waste of time.
But don't fret, he does of course, like many of Dickens' characters, vow to change his life before the story runs out of time.
He is a grave robber.