Hi, and welcome to the family
First, Awasu has its own database of archived content, so depending on your requirements, you may not need to use MySQL. This content is accessible via the Awasu API
If you do need your content in MySQL, there are two options. There is an Awasu extension
available that you attach to the channels whose content you want in your MySQL database - every time the channel updates and new content is received, the extension is run, which transfers the content to your MySQL database. Download the ZIP file in the wiki page linked to above, there is a help file in there.
The disadvantage with this extension is that it was written by one of our old users, who is no longer around, and since he didn't release the source code, we can't provide any support for it. Therefore, it's not really suitable for serious applications. It also need to be attached to each individual channel you want in MySQL, which is tedious if there are a lot of them.
The other option is an alternative extension, that was written by us (so it is
supported), that stores your content in a database (and not just MySQL), but it's not free. Shoot us an email if you're interested in this option.