Short answer

I've tried setting the Application Settings in the portal, both the Connection Strings and the App Settings. And using both "SQLCONNSTR_DefaultConnection" and "Data:DefaultConnection:ConnectionString" as the key.

You're close.

1. Go to Azure web app > configure > connection strings.
2. Add a connection string with the name DefaultConnection.
3. Use Configuration.Get("Data:DefaultConnection:ConnectionString") to access it.

Example using timesheet_db instead of DefaultConnection

This is an example from my own timesheet application. My connection string was named timesheet_db. Just replace all instances of that string with DefaultConnection to adapt the example to your use case.

Azure web app configuration

Azure web app service control manager

The online service control manager at https://myWebAppName.scm.azurewebsites.net/Env will show the connection strings.

Startup.cs

Setup configuration settings in Startup so that the environmental variables overwrite the config.json.

public IConfiguration Configuration { get; set; }
public Startup()
{
    Configuration = new Configuration()
        .AddJsonFile("config.json")
        .AddEnvironmentVariables();    <----- will cascade over config.json
}

Configure the database in Startup.

public void ConfigureServices(IServiceCollection services)
{
    services
        .AddEntityFramework()
        .AddSqlServer()
        .AddDbContext<ProjectContext>(options =>
        {
            var connString =
                Configuration.Get("Data:timesheet_db:ConnectionString");
            options.UseSqlServer(connString);
        });
}

Of course, the example uses a connection string named timesheet_db. For you, replace all instances of it with your own connection string named DefaultConnection and everything will work.

In RC2 I had to change how my connection strings were read to get them to work in Azure. In my case I had to ensure the Azure connection string was named "DefaultConnection", and accessed by:

RC1:

{
    "Data": {
        "DefaultConnection": {
            "ConnectionString": "Server=(localdb)\\MSSQLLocalDB;Database=db;Trusted_Connection=True;"
        }
    }
}

Accessed by:

var conn = Configuration["Data:DefaultConnection:ConnectionString"];

RC2:

{
  "Data": {

  },
  "ConnectionStrings": {
    "DefaultConnection": "Server=(localdb)\\MSSQLLocalDB;Database=db;Trusted_Connection=True;"
  }
}

Accessed by:

var conn = Configuration.GetConnectionString("DefaultConnection");

You have a number of options to set your connection string. The default setup class get the enviroment settings from different sources. You could set your connection string in config.production.json. or config.staging.json. See the startup class

    public Startup(IHostingEnvironment env)
    {
        // Setup configuration sources.
        var configuration = new Configuration()
            .AddJsonFile("config.json")
            .AddJsonFile($"config.{env.EnvironmentName}.json", optional: true);

        if (env.IsEnvironment("Development"))
        {
            // This reads the configuration keys from the secret store.
            // For more details on using the user secret store see http://go.microsoft.com/fwlink/?LinkID=532709
            configuration.AddUserSecrets();
        }
        configuration.AddEnvironmentVariables();
        Configuration = configuration;
    }