"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.

I wasn't using Azure, but I got the same error locally. Using <customErrors mode="Off" /> seemed to have no effect, but checking the Application logs in Event Viewer revealed a warning from ASP.NET which contained all the detail I needed to resolve the issue.

If you add this to your web.config transformation file, you can also set certain publish options to have debugging enabled or disabled:

<system.web>
    <customErrors mode="Off" defaultRedirect="~/Error.aspx" xdt:Transform="Replace"/>
</system.web>