In a local environment it iss possible to connect your application to a Local SQL Server Express Instance.
However when you deploy to Azure you'll need to connect your app to an actual running SQL Server Instance. A SQL Azure Instance is probably your simplest option as you can easily spin one up through the Azure Management Portal. (Though you could also host a regular SQL Server Install on a VM)
Once you've created your SQL Azure Instance, you'll be provided with all the Connection and Auth credentials you need. You'll need to change the DefaultConnection
connection string in your Application Configuration or CSCFG file depending on where your storing it to point at the new SQL Azure Instance, instead of a SQL Express before you deploy your app.
<configuration>
<connectionStrings>
<add name="DefaultConnection"
connectionString="Server=tcp:[serverName].database.windows.net;
Database=myDataBase;
User ID=[LoginForDb]@[serverName];
Password=myPassword;
Trusted_Connection=False;
Encrypt=True;"
providerName="System.Data.SqlClient" />
</connectionStrings>
</configuration>