"As a developer in this new world of cloud applications I want to write a database backed web application and deploy it in the cloud so I can prove my cloud skills and show I can do more than just local development/deployment."
[Setup] You'll need an account on Azure. If you haven't done this before, you get a free account with $200 credit. This is more than plenty to cover this class for free. Do use this resource wisely and remember, if something is provisioned or running you'll have to pay for it. If it takes up storage or bandwidth the cloud provider will charge you for it somehow. Part of this homework and tasks you'll do next term involve learning how to use cloud services properly. Hint: when you're finished with something and don't need it anymore -- delete it. If you'll need it in the future then just make sure it isn't running. Databases cost money just by existing. Delete them if not needed.
Many students last year used their Microsoft Imagine (DreamSpark) account on Azure. They got free databases (otherwise they're $5/month. If you need an account let me know.