I know its possible to use WebSockets within C# using a console application running along side the web application but Im wondering if its possible to use the requests on the C#
Check out the .net client implementations from Kaazing. Using these clients you can connect to any websocket server.
PokeIn 2.0 has built in WebSocket feature and it doesn't require any special environment to work. Details available from here
It's possible to host a WebSockets server within the context of an ASP.NET. However, you'll need to be aware that the ASP.NET application can be restarted which means the server will be shut down and will need to be re-established along with the ASP.NET application.
Hosting the WebSockets application within a separate Windows Service would provide greater reliability in the socket server, but then will require IPC to share data between the ASP.NET application and the Windows Service.
So, there's a trade-off to discuss. It depends on how sensitive your client code is to connection loss and re-establishment. If this is handled sufficiently then hosting within ASP.NET will be fine.
Note that while WebSockets are new to HTML, the concept has existed for many years with Flash and Java Applets. Especially with Flash it's commonly used for online games and web-based chat.
The source code of SuperWebSocket contains a web project which run a socket server in website. http://superwebsocket.codeplex.com/
There will be is built in support for websockets client and server in asp.net 4.5 and IIS8.
See: http://www.asp.net/vnext/overview/aspnet/whats-new#_Toc318097383