I'm using MySQL for the backend.
I know that AWS gives you the ability to set up a free server image for Linux. If you have shell access, you can just use ssh to log onto the server. If not, maybe you can FTP in.
Once there, you can access the AWS management console which should give you server information that you need to add to your scala app.
You will also need tomcat set up on AWS as well. Tomcat should also set up the permissions you need for SSH to work. Also, if you have Tomcat set up, you should be able to use Java 6 or 7 to log into your MySQL database.
I am not a total expert on Play2 or Scala apps, but I have set up several other apps set up this way. I do mostly PHP work and very little Java, so my AWS is set up with mostly PHP.