Hice una clase de solicitud posterior que podría reutilizar para hacer solicitudes POST a una API externa y devolver los objetos que me envían (JSON):

class PostRequest
    {
        private Action<DataUpdateState> Callback;

        public PostRequest(string urlPath, string data, Action<DataUpdateState> callback)
        {
            Callback = callback;

            // form the URI
            UriBuilder fullUri = new UriBuilder(urlPath);
            fullUri.Query = data;

            // initialize a new WebRequest
            HttpWebRequest request = (HttpWebRequest)WebRequest.Create(fullUri.Uri);
            request.Method = "POST";

            // set up the state object for the async request
            DataUpdateState dataState = new DataUpdateState();
            dataState.AsyncRequest = request;

            // start the asynchronous request
            request.BeginGetResponse(new AsyncCallback(HandleResponse),
                dataState);
        }

        private void HandleResponse(IAsyncResult asyncResult)
        {
            // get the state information
            DataUpdateState dataState = (DataUpdateState)asyncResult.AsyncState;
            HttpWebRequest dataRequest = (HttpWebRequest)dataState.AsyncRequest;

            // end the async request
            dataState.AsyncResponse = (HttpWebResponse)dataRequest.EndGetResponse(asyncResult);
            if (dataState.AsyncResponse.StatusCode.ToString() == "OK")
            {
                Callback(dataState); // THIS IS THE LINE YOU SHOULD LOOK AT :)
            }
        }
    }

    public class DataUpdateState
    {
        public HttpWebRequest AsyncRequest { get; set; }
        public HttpWebResponse AsyncResponse { get; set; }
    }
}

el método Callback obtiene el objeto de fecha y hora y lo empuja a esta función:

    public void LoadDashboard( DataUpdateState dataResponse )
    {
        Stream response = dataResponse.AsyncResponse.GetResponseStream();
        //Encoding encode = System.Text.Encoding.GetEncoding("utf-8");
        //StreamReader readStream = new StreamReader(response, encode);
        //readStream.Close();


        Deployment.Current.Dispatcher.BeginInvoke(() => {
            App.RootFrame.Navigate(new Uri("/Interface.xaml", UriKind.RelativeOrAbsolute));
        });
    }

Ahora no estoy seguro de cómo obtener el cuerpo de esa respuesta que me fue enviada desde la API. Está devolviendo un formato json, necesito poder asignarlo a una buena clase de C # y usarlo para mostrar cosas en el teléfono.

No puedo encontrar un ejemplo que no use JSON.NET (que no tiene un ensamblaje para Windows Phone 8)

Este es el error que consigo instalando la clase HTTPClient:

Attempting to resolve dependency 'Microsoft.Bcl (≥ 1.1.3)'.
Attempting to resolve dependency 'Microsoft.Bcl.Build (≥ 1.0.4)'.
Successfully installed 'Microsoft.Bcl.Build 1.0.10'.
Successfully installed 'Microsoft.Bcl 1.1.3'.
Successfully installed 'Microsoft.Net.Http 2.2.13'.
Successfully added 'Microsoft.Bcl.Build 1.0.10' to UnofficialPodio.
Executing script file ***\packages\Microsoft.Bcl.Build.1.0.10\tools\Install.ps1'.
This reference cannot be removed from the project because it is always referenced by the compiler.
This reference cannot be removed from the project because it is always referenced by the compiler.
This reference cannot be removed from the project because it is always referenced by the compiler.
This reference cannot be removed from the project because it is always referenced by the compiler.
Executing script file ***\packages\Microsoft.Bcl.Build.1.0.10\tools\Uninstall.ps1'.
Successfully uninstalled 'Microsoft.Bcl 1.1.3'.
Successfully uninstalled 'Microsoft.Bcl.Build 1.0.10'.
Install failed. Rolling back...
Failed to add reference to 'System.IO'.

"{
    \"access_token\": \"123803120312912j\",
    \"token_type\": \"bearer\",
    \"ref\": {
        \"type\": \"user\",
        \"id\": 123123
    },
    \"expires_in\": 28800,
    \"refresh_token\": \"234234f23f423q432f\"
}"

...

public class Auth
{

    [DataMember(Name = "access_token")]  
    public string AccessToken { get; set; }

    [DataMember(Name = "token_type")]  
    public string TokenType { get; set; }

    [DataMember(Name = "expires_in")]  
    public string ExpiresIn { get; set; }

    [DataMember(Name = "refresh_token")]  
    public string RefreshToken { get; set; }

    //[DataMember(Name = "ref")]  
    //public string Ref { get; set; }

}