How to get Fiddler to show outbound requests such as to the Azure Service APIs:
Add the following into your web.config:
<proxy autoDetect="false" bypassonlocal="false" proxyaddress="http://127.0.0.1:8888" usesystemdefault="false" />
See this post for other ways to set up the proxy but I found this the easiest - http://docs.telerik.com/fiddler/configure-fiddler/tasks/ConfigureDotNETApp
And then enable Fiddler to decrypt HTTPS traffic:
Note: Regardless of other settings, .NET will always bypass the Fiddler proxy for URLs containing localhost. So, rather than using localhost, change your code to refer to the machine name.