The correct way is to have a different branch in your source control for each client. Otherwise you can get into possibility that one client can access another clients page (either by accident, or through malicious intent). By keeping each client in a separate branch, and building that branch for a client specific release, you don't have any cross contamination.
But if you are willing to risk mixing multiple clients, then it's usually best just to have a single .cshtml and rotate around the .css where each .css file is named for a particular client. Also look around into Razor and/or your UI framework's supports for theming. You could have a theme per client.