The critical part in this configuration is network configuration. How are you making sure that Azure Cloud Service machine is able to connect your on-premise machine? Have you configured all the firewall and ports properly to make sure the connectivity is not an issue. With network specific problem you might get general error so you may need to troubleshoot network specific issue by yourself for correctness.
You could use this updated document connecting on-premise SQL Server with Azure Cloud Service. This document uses Azure Connect (which is still on Beta and almost depricated) so you can understand how your network should be visible for your Windows Azure cloud application.
IF you don't want to use On-Premise SQL Server and want to use SQL Azure or Azure Storage then the forms authentication process could be much simpler. Here are more details if you wish to use it: