Are you talking about just making a new forum dedicated to only mafia, or thinking about implementing features that make mafia a part of the website? I feel like you mean the latter, since creating a new forum wouldn't require knowledge of the game.
Basically the game is an un-informed majority (the town) vs an relatively informed minority (the mafia). There is two factions generally with two separate win conditions. As a town player, your goal is to try and guess who the mafia are based on their in-game behavior and actions. Once you've guessed who a mafia member is, it is your goal to convince the as many players as you can to vote to lynch that player since every player gets a vote. Mafia also have a vote too, so you have to have a convincing argument. When the majority of living players have voted a single person and hit a certain lynch number, that person is killed and removed from the game. These lynches occur in what is called the day phase. The night phase follows this, in where there is no public communication. Generally the minority group (mafia) will be the only ones allowed to talk in a private messaging system to each other. Each night phase they have to decide amongst themselves which town member to kill. As mafia you are trying to slowly kill each town member and lead mis-lynches by lying and misleading the town into lynching themselves instead of you and your team. In addition to this, most living players will usually be granted a role or ability that they can use in the night phase to assist them in the game. Both town and mafia are commonly given roles.
That is a brief overview of what mafia is. But to answer your question of how to implement it, I would say a forum works just fine. The only thing that makes it difficult on this website is the lack of a group messaging system, seeing as mafia can't communicate with each other on the site, and we end up having to use third party applications to communicate. If you could implement a chat system, or messaging system that allows for more than 1-1 communication, that would be a huge help to the mafia community!