Sometimes I run games for players who are more interested in backstabbing each other than saving the queen or who would rather take over through world than save it. I invite such players to my table. There is tremendous fun to be had running sessions for a group of would-be villains and it can actually be a lot easier to manage than a standard campaign.
A lot of GMs will tell you that it is more fun to play villainous than heroic NPCs. For me at least that is true. Just like bad guys make movies interesting, they also add to an RPG. For players this can be true as well, some people would rather play an assassin or warlord than a paladin. Provided everyone at the table is on board for this approach, I see nothing wrong with it.
I think there are two basic elements that are present in a Cutthroat Campaign, which are not common in a standard one. The first is a willingness to engage in backstabbing. It doesn't mean characters always kill each other or steal each other's things, but it does mean they will resort to violence against party members when their goals are sufficiently thwarted by them. The second is they are motivated by villainous goals and use villainous means to achieve them. These are not morally upright characters, in many cases they are bad to the bone.
While dropping such a group into a setting creates chaos, it also creates content. It propels the game toward something. Even if they are occasionally bickering among themselves, such characters tend to be driven towards goals that don't require substantial preparation on the part of the GM. There will still be prep and you will still run things like dungeons once in a while, but a lot of what they do is more like a chemical reaction to the environment, like trying to take over local guilds or create a wealth stream by intimidating store owners. Their foes may be other bad guys who want a piece of the same pie or they could be heroes spreading justice through the world.
Some people object to evil player characters or infighting. I can understand this and it isn't for everyone. Others would say it is potentially dangerous or immoral. I don't encounter this much but there are folks who feel this way. My attitude is RPGs are not a more dangerous medium than film or books and you don't become an anti-hero by playing one anymore than Bryan Cranston has descended into a life of crime by playing Walter White. This is make believe. So long as everyone at the table is comfortable with it, I think it can be fun and cathartic for players to take on the roles of villains (whether as mobsters or roving adventurers bent on destruction).
A lot of my games cater to this because it is a style of play I enjoy as a GM. That isn't to say it is how I always run games or how I mostly run them (most of my campaigns are more standard). I just enjoy it when I have the right crowd and the right setting for it. There are some things the GM needs to do though to make this type of game sustainable and fun.
When in-fighting is on the table, being fair is a necessity. By fair, I don't mean give everyone a good outcome or create parity, I mean be as impartial as you can. For me this boils down to being mindful of my decisions and applying the rules equally to everyone. People may quibble over whether true impartiality is ever possible, that is an argument for another day. The point is: make an effort, try, set it as your ideal.
Part of the point of play here is the players will be smashing the scenery. They might bully locals, perhaps attempt to take over institutions or even governments. You shouldn't hand these things to them, nor should you make victory impossible. The setting and its people should put up a fight that matches their resources and abilities. So if the players walk into he halls of some local magnate, play that character to the extent that her resources and skills allow. Sometimes characters can stroll in and take things over, other times they will be in a position of extreme disadvantage and have to submit to the will of a greater foe.
If players walk into a town and start beating people up in the streets for no apparent reason (a possibility in this type of campaign) there people are likely to react. You need to think about what that might be, because it won't always be the same (and if it is, that gets dull quickly). Some people might run, some might try to appease, and others might resist or attempt to bargain. Occasionally someone with the means and character will step in and try to teach them a lesson, possibly even attempt to reform them.
Communicate and Know the Boundaries
Communication with your players is important in this type of game. You want to make sure everyone is on board with the approach and that no one is doing anything that makes them uncomfortable. So ask what they're okay with. If you sense someone is troubled but isn't saying anything, take time to talk to them after the game or before the next session.
The above advice isn't fully comprehensive. There are a lot of details that I probably over looked. I will likely revisit it in a follow-up as I am presently running another Cutthroat Campaign.