Try this, but you will lose all files etc that you have not backed up. If you are using the same computer it will restore from back up.
1) plug your phone to your computer while running iTunes.
2) perform hard reset by holding home and sleep as you've done before
3) wait for phone to reset Apple logo will go off and come back on... do not release buttons!
4) keep holding until the phone resets one more time and displays the recovery mode screen
5) follow the instruction for 1) restoring from backup or 2) setup as a new iphone
If that will not resolve your problens, it may be time to take it apart and have a look at the obvious parts that could have failed. Unless you have it apart, anything will only be a guess. Start with the battery if you must. Hope this helps, good luck.