Jose is Cole's fellow inmate and is later conscripted—sorry, "volunteered"—to become a time-traveler. His job, like Cole's, is to find information that could lead the Scientists to find a pure form of the virus.
Also like Cole, his job is the pits.
He appears briefly in the World War I trenches, screaming, "I gotta find 'em! I gotta find 'em!" Presumably, the "'em" in question is the Army of the 12 Monkeys. His fate remains a mystery until late in the film, when he reappears at the airport, looking way worse for wear thanks to his time-traveling adventures.
At the airport, he delivers a message to Cole and that message comes with a free gun. He tells Cole to shoot the man who will release the virus, or Jose will have to shoot Railly. Cole initially tries to choke out Jose but then notices the Geologist in the crowd, and Jose admits,
JOSE: You see? I had no choice. These are my orders, man.
This scene ties nicely into the film's themes of power and determinism. Again, like Cole, Jose lacks the power to make his own decisions. He's just a pawn on the board, a cog in the machine, a…um, other cliché that means the same thing.
Unlike Cole, he doesn't try to fight it. He accepts his role and does what he needs to do to survive. Although we don't know his ultimate fate, we're guessing it ended with fewer bullets in the back than Cole's.