Carmine is a Rottweiler and loyal friend to Christopher Chance. Chance has owned him since the last mission of his life as an assassin, which sent him in his current direction to help people. He is about 6 or 7 years old in Season One. The dog is likely named after Carmine Infantino, one of the two creators of the comic the series is based on.
Carmine was originally owned (as an adorable puppy) by Katherine Walters, an operations administrator for the Port of San Francisco who Chance was assigned to kill, in what would prove to be his final mission as an assassin. Chance, known at the time as Junior, was first introduced to the dog when he infiltrated the San Francisco Police Department safehouse where Katherine was being kept for protection. Since he was posing as an assistant U.S. District Attorney, Katherine was surprised that Carmine took to Junior, noting that he was not fond of lawyers.
When Junior determined that Katherine was not the subject he was led to believe she was, he helped her escape police custody to protect her, taking Carmine with them. The trio eventually ended up at the warehouse office/home of a former bodyguard named Christopher Chance. While Junior and Katherine went to investigate information regarding a ship named Leonor at the port, Carmine stayed at the bodyguard's place.
Katherine and Christopher Chance were both killed at the dock by an assassin named Baptiste that Junior had trained and worked with. Junior, having gotten out of his mercenary lifestyle, returned to the office/home and claimed it as his own. He found the young pup, now as alone as he was, and decided to keep him, most likely as a reminder of Katherine and a symbol of his new life. Carmine often accompanies him on runs.
Carmine does not appear in the refurbished HQ provided by Ilsa Pucci, although he is in the first episode of season two, "Ilsa Pucci", at the start. None of the characters, not even Chance, comment on or explain his absence.