Here is an article about setting up AWS EC2 to run Play and Mongo: http://stevenwilliamalexander.wordpress.com/2012/06/10/amazon-ec2-mongo-play-scala-instance-setup-gu/ and there are also guides for binding to Amazon Simple DB and Dynamo in Scala as well that might be much easier setup than using MySQL.
I've used a lot of AWS services for building composite services and I'm available for a call if you get stuck or need any advice on best practices.
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.
Heroku is an easier starting point for a lot of people, but if you want to go with AWS, this might help get you started: