I think the most standard solution is to have your C# service export a "web services" remote API, which the C code can then access via the standard Linux web services packages. In effect these support a form of RPC, but with web pages encoding the request and also the reply. The advantage of using web services is generality; your solution will port to any platform and will work from any language, not just C.
Another way to go is to use one of the new open-source packages from Google, Facebook, Twitter, etc. A bunch of the cloud vendors are open-sourcing their RPC infrastructures because many customers pose your exact question and need these solutions. Those will definitely be faster and more powerful in other ways too: they include powerful performance visualization tools and debuggers of various kinds. Microsoft has a remote method invocation technology too. So these are an option, perhaps less general but definitely faster. So you then have to ask how important speed is, for this particular path...
The bottom line then is that there are maybe five packages you could use. No idea which is winning or best!