You might want to look into a class of software called Interactive Fiction. I actually spent all of Saturday reading up on it as I was looking at different approaches to dialog/conversation systems in games for a project I'm working on. This kind of game is more writing-intense than programming...but if you want a pure story, there are some great tool kits out there that can carry much of the programming weight for you. RenPy is an example of that.

Indigo Prophecy and L.A. Noire are some other games you might consider; the former has an interesting approach to your interrogations (you have a time limit on considering what questions to ask and actually interject with) and the latter has its "interrogation mode" which people seemed to enjoy. (I actually own L.A. Noire but haven't gotten a chance to sit down with it yet.) Youtube some gameplay from each one to see what I mean.

In general, if I wanted to make a CSI game (and I wasn't concerned with target demographic) I'd focus on the convo system, questioning witnesses and interrogating perps, and a third person FPS element where cover mechanics were huge. Other aspects of the game I'd either eschew or make as mini games, like searching crime scenes for evidence and the analysis of that evidence. But to be really true to a show like Criminal Minds, you'd want it to be mostly character development which you draw the player into, and that would benefit most from some ambient music and a very deep, content-loaded convo system...in my mind, anyway.

I hope that's helpful.

I have a lot of thoughts on it. I don't know what the market would be for a game that featured CSI show like characters and also shooting. I had thought that the target demographic for CSI shows were like 40+ people who enjoy TV for their after work decompression...as opposed to video games. That's not a market you are going to sell on a game with shooting elements, I think. (I have ZERO research to back that up; it's just my gut.)

Were I you, I'd go for the meat of it first with a prototype. Write up a very short story with some branches (a choose-your-own adventure kind of thing) and go interactive fiction with it. Do a single interrogation where the player can, I'm just making stuff up here...

• appeal to the suspect by going buddy/buddy with him;
• put pressure on the suspect;
• ask a set of questions.

Before the player interrogates, a brief conversation with an associate and/or a review of their "in-game notes" can prime them, maybe present the evidence they already have that they can use to spring a question on the perp that they already know the answer to. That's a pretty big element in shows like that: catching the suspect in a lie puts the interrogator at a huge advantage in those shows.

Were I you, I'd spin up a prototype to see how it felt and help me decide where to go from there. I'd also play every game out there even remotely like the idea to get a feel for the environment and build up the questions I'd need to ask myself about design and development. I'd also watch a TON of CSI episodes and take notes about the interrogation scenes.