You can use Fleck to create a WebSocket server in C#. Here‘s a VS2012 solution.
The you can use this jsFiddle to run the client.
To learn the Client Side read this article.
You can even test this on a mobile device here
For running on mobile you need to put you mobile device in the same WiFi network as the server then change the socket address in the client from localhost to whatever IP the server has.
Have fun!
PS: Mobile test requires jsFiddle account.
