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.