You don't need to run Javascript inside your C# code. .NET has its own set of functionality to get the information.
The System.Net.WebClient class has a .DownloadString() method that you can use to get the same information from the Skype API as you would in your JavaScript code.
Take that string that you receive from there and use Newtonsoft's JSON DLL (available through NuGet) to turn that string into a JSON object that you can get all of the information from.
Here is an example of the code that I used to do this in one of my projects, modified for generality. For reference, the DLL's you'll need to add usings for are System.Net, and Newtonsoft.Json.Linq.
private static GetData(string urlParameter) {
string response = string.Empty; // This will be the data that is returned.
string url = "http://skype.api.com/someCall?param={0}"; // The API URL you want to call.
using (var client = new WebClient()) {
// This will get the data from your API call.
response = client.DownloadString(string.Format(url, urlParameter));
}
JObject obj = null; // This is the Newtonsoft object.
try {
obj = JObject.Parse(response);
// Continue on processing the data as need be.
}
catch {
// Always be prepared for exceptions
}