The DeLuca Family Immediately comes to mind, MC is recruited into a mafia family and essentially works his way up.
DropOut Saga I didn't really get into, but it's got a pretty big Yakuza/Gang theme going from what I remember. The MC was part of one and wanted to get out of it.
Lord King kind of fits, you become a faction leader in a post-apocalyptic city after taking out the previous one.
Pretty sure there's others I'm forgetting. There's also a fair amount of games that have mafia/cartel/gangs in them but most of the time not as a major focus.