For 500 errors, one good way is to turn diagnostics on and to look at the logs - both the application event logs and the failed request logs might help. Here's one post about this - http://oakleafblog.blogspot.com/2010/11/adding-trace-event-counter-and-error.html
As Igorek has said, you can also use RDP (remote desktop) to log in and to check the app - here's a good blog post about this - https://blogs.msdn.com/b/jimoneil/archive/2011/04/11/500-and-other-errors-in-azure-deployments.aspx
My guess is that you might be missing one or more Assemblies from the deployed package
