I've been running into the same problem, just registered to let you know. It seems that it isn't possible right now (who knows why, but I'm looking at you, ASP.NET RC1) to publish while deploying an SQL server. However, it DOES work to publish without one and make one in the portal, and then connect them through your Web.config.
At first I thought it was all of the new updates to Visual Studio (as of 5/12/15), but I was still getting the same problems after updating. I'm not sure if the new updates are required for this workaround, but that's what I was working with for this.
Steps:
1. Publish your API app as normal, without including an SQL server.
2. Go to your Azure portal and navigate to the "SQL Databases" tab.
3. Create a new database using your existing server, or just make a new database server if you don't have one yet.
4. When it's done (it may take a second for it to show up in the list), click on the new database and it's settings will appear. You want to go to the "Show Database Connection Strings" option.
5. Save the ADO.NET connection string!
6. Go back to your project in Visual Studio and navigate to the Web.config file.
7. In the "connectionStrings" header, find your database context. It should have a name formatted like YOURDATABASENAMEContext, or something similar. It'll probably be the second entry there. Replace (save a copy first, see note below for why) the inside of the "connectionString" with the ADO.NET connection string of the database you just created.
8. Publish and it should work!
NOTE: This workaround, if followed literally, means the program won't function locally anymore (e.g. starting the app rather than publishing it) since it will be looking for that database in the connection strings. You could return it to it's original state for testing, and then use the azure connection string when publishing, although I'm sure there's a more elegant way to do it.
