my first suggestion would be to check the dock connector. make sure that there are no broken or bent pins, no corrosion and no foreign matter in the connector. Then take is apart and make sure that your baatery connection is not corroded or otherwise damaged. Change the battery as well. Careful when you remove the battery, the connector has the tendency of getting broken of the board. Use a paperclip and bend it into an L-shape and use the L to pull stright up. Make sure that your logic board has no crrosion on it. If that still wont work use a multimeter and measure the voltage on your dock connector on the logic board. The schematic is available at pinouts.ru If none of this should resolve it, a new logic board may the only solution. Hope this helps to get you started. Good Luck.
iPod Classic Thin Replacement Battery
iPod Classic 80 GB and 160 GB Thick Logic Board
iPod Classic Repair