The series is very unclear about this. It seems to be some sort of state-sanctioned gang, conducting murders and running brothels - at one point they're tasked by Augustus and Mark Antony with killing a whole bunch of men that could warn Brutus of their alliance. The series takes a lot of liberties with history so it might not have been a real thing, but I'm curious to see if it was. The leader is referred to as a Captain, but I think in Latin that just means 'head' or 'leader'.
It was actually. Rome, in those times, actually had no police force. It was also illegal to bring active soldiers into the city under arms.
During the political reforms of the later stages of the republic, physical violence and force became common tactics used by different sects in order to suppress opposition and enforce their own agendas. In the show Rome you can see an example of this when Marc Antony is trying to get to the Senate to veto and is attacked by a mob of club wielding thugs. Those thugs are meant to intimidate and threaten participants in the forum to promote a political agenda.
Senators and Consuls eventually clue up to the fact that having armed gangs full of guys like that running around the city bullying your political enemies was a great way to make sure you win elections and stay in control.
Of course that would not be without some backlash for the prestigious and eventually the oligarchy turns to organizing the gangs under proxies. Two of the first and most famous are Milo and Clodius.
Here is paper further explain their activities within the context of mob warfare in Rome with a bit of editorializing. There is a bibliography at the bottom with further source material if anyone is interested. A few sources, particularly Mob Violence in the Late Roman Republic: 133-49 B.C by J.W Heaton and Violence in Republican Rome by AW Linott would be relevant.
Great comment by /r/AFancyLittleCupcake
Also, when you they're tasked with killing people on the "list," that wasn't actually murder, it was state sanctioned "proscription." Like the original concept of outlawing someone, it was the state declaring "hands off, you can totally kill this guy if you want." They refer to this a few times in the show.
Also, there have been several threads here before about the historical accuracy of the show, which you might find instructive.
This is slightly off topic but I thought the character of Julius Caesar was generally unlikeable with zero charisma. Being such a central figure in the story, this almost ruined the entire series for me.
Does anyone care to comment on this?