The issue can be one of 3 things.
1. the charger doesn't work properly anymore - if possible try another charger. Don't buy a new charger, if possible borrow or take it to someone to test. --- If the phone works with the charger but the battery wont hold charge, then the charger is fine go to 2.
2. the battery is dead. And would need to be replaced. this is the most likely problem.
To replace the battery, follow these instructions:
you can buy a new battery from http://www.directfix.com/mm5/merchant.mvc?Screen=PROD&Store_Code=GHT&Product_Code=PA1547&Category_Code=_TREO
if this does not fix the problem then. the problem could be the charger (1) or most likely the mainboard (3)
3. the phones mainboard is faulty and its unable to charge the battery.
You could try doing a hard reset, but its unlikely to help, the phone probably needs repairs. (A hard reset will delete everything on your device.)
to perform a hard reset, if you need instructions please click open this page and follow the steps under treo 600 http://kb.palm.com/wps/portal/kb/common/article/42094_en.html#treo600
if the problem remains after replacing the battery and trying another charger even and hard reset, then the phone is most likely in need of repair.