Moved static field initialization to static constructor and performed some ReSharper formatting
Increase the detail provided by WebClientException.Message by including the URI and the integer value of the StatusCode.
Remove the request and response fields since they aren't really used.
Build WebClientException with a StringBuilder to make each step simpler to follow and allow for more variations over the different scenarios of having request, response and/or body.
Console.WriteLine of the WebClientException to see how beautiful it is.