Did you already try to set it up "as new device"? Try to connect in recovery mode first.
Recovery mode is explained here: iPhone and iPod touch: Unable to restore
Setting up "as new device" here: How to back up your data and set up as a new device
Don't use the backup after you set it up again, in case it contains damaged files causing your problem.
If the screen does not come back after restoring, get it serviced by Apple.