The web service being implemented in C# shouldn't mean anything to the client (applet). I would start by downloading
Axis, throwing wsdl2java at the WSDL document of the web service, and including the generated classes into your applet's .jar file (and whatever libraries the Axis client requires).
It really doesn't matter that it's an applet you're writing instead of a standalone application or a server-side web application. Any web services client tutorial will do.