I think the video game style that fits the muscial Heathers and the characters the best would be an RPG game. RPG is short for Roleplaying game, and it's usually a game with a fighting system that has a story that goes along with it. One of my favorite examples of an RPG game would be Omori. Omori is an RPG game that follows a boy hiding from the truth in his dreams. The reason I think a RPG would be perfect for Heathers is because, it already follows a story, and the characters do fight a bit, like physically. It would be hard to loose the game because there would be one path leading to the main ending.
I'd keep the characters looking like how they did in the musical. The Heathers (+ Veronica) had color coded outfits, you could tell them apart that way. I really liked that idea! For the other characters like JD I'd probably keep his pretty similar too. ]
For the most part theyre at a school, so i'd most likely make the background a locker bay or a place thats like the student lobby. Some other important locations are the park and Heather Chandlers house. I say those places because that is where people died.