Is this easy to do? I think I can make a AMI, ...
I think the answer depends on how comfortable you are with system administration in general. Creating a AMI to run in EC2 is really pretty much the same as creating a physical server or a VM image. You'll need to install an operating system, and then install tools, libraries and programs you need (like mysql, the jdk, ssh, etc).
You can save yourself a little work by using one of Amazon's pre-built AMI's http://aws.amazon.com/amis/. But, ultimately, you'll be responsible for all system administration of the server. If you've never built a server from the ground up, you have a pretty big learning curve ahead of you. It's not insurmountable, but just be warned that the devil is in the details; there's a ton of stuff you'll need to learn ;-)
... but I'm not sure how to upload Java files, compile and run them, ...
Once the server is setup and running in EC2, compiling them and running java files is just the same as compiling and running on your local. Normally, you probably want to compile and package your java app into a jar or war and then transfer that up to your EC2 server. If you install linux os on your EC2 server, you can use scp or a FTP client to transfer your files over sftp to move the files from your local up to the server. Once the latest files are up on your server, you can ssh to the server and start your app.
... and create a MySQL database etc ...
Installing mysql is going to be specific to the OS you choose to install on your server. For example, you can install mysql easily on Ubuntu with a command like:
sudo aptitude install mysql
Again, there will be more system-admin-type stuff to learn here specific to mysql databases.
So, it's definitely doable. An experienced sys admin could build a AMI instance pretty easily/quickly. If this is your first experience with system administration, I'd suggest finding an old Desktop you have lying around and try installing Ubuntu and all the required libraries and tools you need (mysql, jdk, ssh, etc..). Get your java program working on the old desktop and then it should be pretty easy to create an AMI from that. Then you can run your custom AMI on EC2 and will be set up.
If you don't have a spare desktop lying around, you can use one of the Virtual Machine products like VMWare Player or Sun's VirtualBox and build a server instance on one of those.
If you want to avoid the hassle of managing the entire install of the Operating system, you might want to look at services like slicehost and/or linode instead of EC2. They give you ssh access to a pre-installed server. And it's as easy as clicking a button to install programs like mysql, etc.
Hope this answer is relevant and helpful, good luck. - Dave