migrate db's - best way to mirror or replicate a full database
It sounds like you're trying to setup a disaster recovery environment, so that if your local machine blows-up you can switch over to the Azure machine.

There are a number of options, each with their own pros and cons. These are well documented elsewhere (see this link for an overview from Brent Ozar: http://www.brentozar.com/archive/2011/12/sql-server-high-availability-disaster-recovery-basics-webcast/). That said, you might want to look at Log Shipping and Asynchronous Database Mirroring particularly. 

Log shipping is an automated process for taking your log backups and restoring them on another server (e.g. Azure), you can take log backups every five minutes so that you're Azure server is always within about five minutes of your local machine. The upside is that it's easy to configure and can be setup through the SSMS GUI.

Database Mirroring takes this to the next level, constantly sending completed transactions to the other server (i.e. Azure) so that it's within a minute or so of your local machine (this depends on your network connection, of course).

There are other options but if you're new to disaster recovery but I'd recommend looking at these to start with.