It could be a couple things. The charger may be faulty due to a bad transformer or a bad wire. they break easily. The phone itself may have a charging port that has failed, or is filled with dust to the point that the charger won't actually connect, and while it was plugged in long enough to be fully charged, it may in fact be completely dead. Or it could be the phone itself or the memory card in the phone is bad. Was the phone dropped at all? it could be that the sim card or the sd memory card if it has one got knocked partially lose and the phone won't restart because it is in error state.
If it's two years old and has known wierd screen issues, I'd just go replace it personally rather than screwing with it. You can probably get a better phone free or close to free at this point regardless of what cell contract you are on.