Gaius and the Senator Falco were both introduced to General Maximus Decimus Meridius by Commodus. When Commodus became Emperor, Gaius was a major leader in the government of the Roman Empire. He seemed to dislike the fact that for 150 days straight Commodus had games in Rome. He met with Gracchus and discussed the oddness of the games and their lenghthy timelines.
It is implied, through his strong association with Gracchus rather than Falco, that he supports the formation of a Republic. The viewer is also left to presume this, after Commadus has him killed, by having a (presumeably poisonous) snake put into his bed as he sleeps with his mistress.
Gaius and his fellow senator Gracchus are a play on Gaius Gracchus, a real Roman senator.