IPod wont turn on or sync to computer to charge

Connect the iPod to different computers. If it doesn't even go to the "do not disconnect" screen on any of the computers, you have a broken logic board. You can try to replace the part yourself, it is available on any auction website. There are iPod touch Repair services that can fix it for you, but if you're a little tech savy you should use an iPod touch repair guide that teaches you how to do the repair yourself.

Your battery is fu.k.d Take the thing back to the Apple store or Best Buy to get a free replacement if your within your year warranty.  One caveat, if you mention anything about dropping this thing or exposing it to humidity or water, then they will void your warranty and won't give you sh.t