“An exception occurred while processing your request. Additionally, another exception occurred while executing the custom error page…”

First, set customErrors = “Off” in the web.config and redeploy to get a more detailed error message that will help us diagnose the problem. You could also RDP into the instance and browse to the site from IIS locally to view the errors.

<system.web>
      <customErrors mode="Off" />

First guess though – you have some references (most likely Azure SDK references) that are not set to Copy Local = true. So, all your dependencies are not getting deployed.

Get to the detailed error first and update your question.

UPDATE: A second option now available in VS2013 is Remote Debugging a Cloud Service or Virtual Machine.