Make sure it is off and take a toothbrush and dip it in rubbing alcohol (I usually wipe in a paper towel before I go any further to make sure it is not soaking wet). Then brush the charging port. Let dry, then plug in and see what happens.
Have you tried a different power cable?
You can also try to factory restore it. Press and hold the menu button and center button and the same time until you see the itunes logo screen. Plug into a computer with iTunes and restore the device - this will ERASE all data on device