If you want to use networking and your own database in the VM, it is best to start with reserved websites since shared websites may not be good idea due to resource limitation and other restrictions.
If your DB is MongoDB, then you have two option:
Have your ASP.NET MVC application running in a Virtual Machine (Windows Server + IIS + your app - IaaS) and have MongoDB is running on other Virtual Machine. If both machines are in same DC, you dont need to fiddle with networking.
Have Azure ASP.NET MVC Web Role (PaaS) and a Virtual Machine have MongoDB is running (IaaS).
In both cases you just need to configure a connection string to connect MongoDB and there is no extra network modification needed to make it work as all machines are inside same DC.