He is looking for a caravaneer named Ringo, who has taken refuge in Goodsprings. Cobb is known as an escaped convict of the NCR Correctional Facility and if a Speech check is passed he will tell the Courier that he was put in the facility for crimes of robbery, arson, and murder.
If the Courier has already agreed to help Ringo battle Joe Cobb and his powder gangers, Cobb will refuse the request to help him take over the town, stating, "No. There's something not quite right about you."
If Run Goodsprings Run is completed, Joe Cobb will warn the Courier that some powder gangers will be hostile no matter what. He will also recommend that the he/she goes down to the NCR Correctional Facility, advising to mention Joe Cobb's name. When the Courier arrives at the NCRCF however, none of the non-player characters have a dialogue option to do so.
During the quest Run Goodsprings Run, when talking to a random Powder Ganger, one might say "I got dibs on the old doc's house." When the quest is completed, Joe Cobb is instead living in Doc Mitchell's house if he lived through the fight with the town militia.
Killing Cobb before the battle in Ghost Town Gunfight, several characters will remark on his death. Trudy will worry that Cobb's cronies will be out for revenge, Sunny Smiles will hope that he was killed for a good reason and not for pleasure, and Ringo will state that his death doesn't mean the powder gangers will go away.