You may not have heard, but you can use Azure's KUDU UI interface to help you diagnose and debug your Azure Web Site. In order to access KUDU interface simply navigate to your Azure Web Site and specify "scm" as part of the address. Such as https://mysite.scm
NOTE: You should be using https to protect any sensitive information exposed by the KUDU interface. You will also need to enter your deployment credentials when prompted.
You can also use a CMD or ps1 interface. NOTE: The CMD interface is a full fidelity wrapper over cmd.exe (Thanks to David Ebbo
@Microsoft for letting me know)
There is also a process explorer to gleam insight.
Another helpful tool is that you can get access to the log files immediately from the 'Diagnostic dump' menu.
There are even more things you can do such as create directories, read app settings, determine the server's configured HTTP header, just to name a few.
You can already see how this interface helps with debugging Azure Web Sites, so get out there and try it out!