Plug your phone into the charger that came with it and plug that into a known good wall outlet. Let it stay plugged in for at least an hour. Ten try turning it on.
If it doesn't come on reset your device. Press and hold the Home and Sleep buttons simultaneously ignoring the red slider should one appear until the Apple logo appears. Let go of the buttons and let the device restart. See if that fixes your problem.
If the phone comes back on leave it connected to the charger until the battery shows 100%.
If the phone still does not come back on make an appointment at an Apple Store to have your phone checked out by a technician.