This guys is the most unsuccessful Jack of all trades on the planet.
He starts off as an influential art dealer, then becomes a pottery manufacturer, and finally gets into the business of selling religious objects—anything to make a buck. When Frederick first meets him, Arnoux has the world at his feet:
He was republican in his opinions. He had travelled; he was familiar with the inner life of theatres, restaurants, and newspapers, and knew all the theatrical celebrities, whom he called by their Christian names. Frederick told him confidentially about his projects; and the elder man took an encouraging view of them. (1.1.40)
Arnoux immediately takes Frederick under his wing. In fact, Arnoux is sort of a model for Frederick's life plan: part father figure and part mentor in romantic relations. This is the guy who got the girl, after all.
Madame Arnoux is Frederick's dream woman, but Arnoux consistently cheats on her with Rosanette. Oh, and he pays off Rosanette when she's upset. And remember that one particularly messy scenario he gets into with the cashmere scarf? Yeah, turns out this guy isn't as great a role model as we might have thought. To top it off, his business practices are pretty slimy, and he ends up driving his family into the ground financially.
Over the course of the novel, Arnoux degrades everything: artistically, financially, and personally. And for one last huzzah, he never pays back the money Frederick loaned him (which, by the way, saved him from total disgrace), and takes off into the countryside for 16 years. Good riddance, dear ol' Arnoux.