Mike, the first thing that I would do is to replace the battery. You can of course open it and check the connection of the battery and see if it is on. Also pay close attention to the actual connector, the 5th gen has an issue with the connector coming of the logic board :) step 5 of the guide. Installing iPod 5th Generation (Video) Battery good luck and let us know what happened.
You may wish to try plugging it into your computer, opening up the Disk Utility app, and see if it shows up in the left hand pane. Go from there - it may be a software issue. My 80 GB 5th Generation iPod Classic showed a dead screen, and now I'm getting a faint 'Please wait - Very low battery' issue on the screen, so it's probably my battery, although I've basically only used it as a storage device - go figure.
Common error amongst these iPods, the magic trick is getting it into service mode. Youtube videos are plenty online: http://www.youtube.com/watch?v=SsWgCLAvt...
Basically connect iPod to Mac / PC, hold menu + enter till restart, then do the same again but before the apple logo (with black background, not the faded grey) quickly switch and hold menu + backward (left), until the service mode appears.
Run all the tests automatically (note: some require interaction), when service test is done, just restart again (menu + enter) while connected to Mac / PC, and wait for it to get recognized by the computer.
(This is especially useful, once the battery is sort of completely empty, doesn't matter how long the iPod has been laying in the dust while turned on.)