Availability sets are limited to Virtual Machines within a single cloud service. Even if you added your database server in the same cloud service as your app servers, this still wouldn't provide HA for your DB since you can have the host OS updated, bringing your DB server down for a short time.
The only way to have HA is to run multiple DB servers. If you run SQL Server, you can set up an Always On configuration (see here for documentation). EDIT I missed the comment about you using MySQL. MySQL has HA configurations, but I haven't set one up before; you'd need to set this up across multiple VMs, and you should consider putting all nodes in an Availability Set, so that any Host OS updates will be staggered across the VMs, rather than being applied at once (and also separating the VMs across different fault domains).