Add this to the system.net section in your web.config
<proxy proxyaddress="http://127.0.0.1:8888" />
127.0.0.1:8888 is the default address that Charles/Fiddler uses.
If you are doing requests against a site that uses HTTPS you can get some certificate errors like this one:
The underlying connection was closed: Could not establish trust relationship for the SSL/TLS secure channel.
Just add this line of code right before you are executing your WebClient requests
ServicePointManager.ServerCertificateValidationCallback += (sender, certificate, chain, sslPolicyErrors) => true;
You could also add the following to your web.config
<servicePointManager checkCertificateName="false" checkCertificateRevocationList="false" />