How do crime organizations spread their influence?

I'd read up at the FBI site about organized crime.
https://www.fbi.gov/about-us/investigate/organizedcrime

Also, try searching for sociology and organized crime, that may get into systemic details around it.

My take: in general you can view it from a simple economic view around supply and demand. You also factor in profit and risk. Crime inherently embeds a risk/reward mechanism where risks are miscalculated along the lines of gambler fallacies.

Organization of crime alters the perception of risk as well as distorting rewards and introducing social rewards in addition to material rewards. The value of the social rewards can be contrasted by the availability of other sources of social rewards and membership.

Strong social communities thus inherently resist organized crime by providing support and offering lower risk social rewards. Weak social communities typically have transient populations, absentee landlords, and higher unemployment rates, though it gets a bit "chicken and the egg."

My suggestion would be to research TV and/or movies since you're really trying to get the "feel" of organized crime without really giving step-by-step instruction on how to build an empire.

Things I would look at:

-Godfather movies
-Daredevil season 1 (on Netflix)
-Goodfellas (movie)
-Untouchables (movie)

That should be a good start.

I would suggest that, for a game, the feel you should be going for is the Hollywood perception of how organized crime works, as opposed to how it really works. Reality is both too messy and too banal to make for a fun game.