A family of Microsoft relational database management systems designed for ease of use.
In Visio Professional (and with 2010, also in Premium), there are two separate solutions for doing Database schema diagrams and UML model diagrams. However, they are not able to work together in the way that you describe. If you want to create a comprehensive Database diagram using the Reverse Engineering capability of the "Database Model Diagram", you could possibly create your own shapes to represent the classes and essentially annotate the schema diagram. But this would not be easy, and neither the DB model solution or the UML model solution have public APIs, so you can't automate either one.
There are likely other modeling tools that will get you much closer than Visio will out of the box. Just by searching on "Model database and C# classes", I found several sites that you can scan for more information :http://www.bing.com/search?q=model+database+and+C%23+classes&form=QBRE&qs=n&sk=
If you really want to have a go at creating your own solution, I suggest you start with the Visio SDK. For Visio 2010, the SDK can be viewed / download starting from here :http://msdn.microsoft.com/en-us/library/ff758690.aspx
Barb