can someone explain to me how to establish an RCON connection to a server with C#?
Do I need a webrcon server to send and receive data or does it all go through the websocket library?

Google: C# server rcon

First result - good C# implementation
https://github.com/stefanodriussi/rconsharp


Second result -Facepunch implementation but in web, you still can understand what for what https://github.com/Facepunch/webrcon/tree/gh-pages

Try having a look at websocketsharp. Then for your address use "ws://ip:port/rconpassword"