Before you start: if you are on Windows, check out these instructions. Most of what is below applies to Linux.
To install eMusic/J you will need to download it. Once you have done this, simply extract the file you downloaded somewhere. Anywhere will do, just remember where it was. After extracting it, browse to the emusicj-linux directory that was created, and run the program in there called emusicj. After a second or two, the program should come up. If it does, all is well.
There is a handy how-to for setting it up on Ubuntu available on the Ubuntu forums.
Note that you will need an implementation of Java 1.5 or better to run this. Here are instructions for installing Sun Java.
If the program window doesn't appear at all (give it a few minutes, just in case it's being slow), then run it from within a terminal and see what diagnostic messages come up. Have a look at the troubleshooting page to see if anything there helps. If you can't work out what is going on from this, then feel free to get in touch with me and I'll try to sort it out.