Bruno Kirby makes a point of telling the story of how King Benny waited 8 years to seek revenge on an assailant in his younger years. By the time Shakes goes to work for King Benny, he's an old man. Obviously involved in organized crime. Shakes gets a job running payoffs to the cops for Benny. But do they ever specifically say what those payoffs are for, or what Benny does?
You've pretty much summed up all the information we know.
In the novel, there's a few extra bits of info, but not much. You've already discussed King Benny's wait for revenge when he was younger. Lorenzo does say this in the book:
When my friends and I were young... Hell's Kitchen was run by a man named King Benny. When he was young, King Benny was a hit man for Lucky Luciano.
So he was originally a lowly hitman for a mob boss.
Later, when Lorenzo gets his first job he obviously started to bring payments to the police:
My first job for King Benny paid $25 a week and ate up only minutes of my time. In the dark club room, one of the guys would hand me a crumpled paper bag and direct me to one of the two local police precincts for delivery. It was a perfect way to handle payouts.
As to what these payments were for? Well, it's never stated in the movie or book. I think it's fairly logical to guess that they were to turn a blind eye to certain mobster activities he was conducting, or to possibly harass a member of a rival faction. It's difficult to know how much they bought him, but I'd say the former is pretty much a certainty.
As to what he actually does however, there's no real info given. It's obvious he's the new mob boss, but what that entails (i.e. drug trafficking, gambling etc) we don't know and don't get told in either book or movie. Whether it's relevant or not I don't know, but the movie is set in the 1960s, which was the same era as the Sicilian Mafia trials regarding growing involvement in heroin trade. That could be his main criminal activity, but it's impossible to know for sure.
Sorry bud, but in the movie/book, they make it a point to mention that drugs and dealers weren't allowed in hell's kitchen with the murder/lynching of a drug dealer.
Furthermore being a hit man for lucky Luciano would mean he was more than likely a higher up in the mafia than just your average hitman, considering he is now running hell's kitchen.
I'd say he was more into the traditional crimes that involved the mob i.e gambling, women, extortion, money lending and other things of that nature.