The key to doing this is a small project called AMFPHP. As stated on their webpage, the project was originally started by Wolfgang Hamann. The team has grown lately, and now has about 5 or 6 developers. Thanks to their hard work, we can now have a Flex based front end to our PHP backend applications.
He creates another application from scratch, providing all of the database structure, XML, and ActionScript, and PHP code that you'll need to follow along. The application is a simple display of database records, in this case user data from the MySQL table.