Phone is off / black when I plug it into charger, only the Apple logo is displayed

Give TinyUmbrella a try. It's a powerful tool to exit the recovery state of the iPhone without the data loss.

1. Make sure that your iPhone is on (showing the Apple logo in your case)
2. Connect your iPhone to your computer via USB
3. Download and install TinyUmbrella
4. Launch TinyUmbrella and wait until it recognises your iPhone
5. As iPhone gets recognised press exit recovery shown in the main menu

In most cases, you can solve these problems by putting the phone into recovery mode and restoring it from backup. To do that, follow these instructions:

1. Connect the iPhone's USB cable to your computer, but not your iPhone.

2. Turn your iPhone off. If the phone won't turn off in the traditional manner, just keep holding the buttons until the screen goes dark.

3. Once you've turned the device off, press the home button on the face of the device and plug in the USB cable to the device's dock connector.

4. When it turns on, keep pressing the home button until the iTunes logo appears on the screen.

5. You'll know the phone is now in recovery mode because a window in iTunes will tell you.

6. You can then start the restore process by clicking the "Restore" button in the iTunes window.

If this still doesn't work, then you can use commercial software such as Dr.Fone to fix this kind of issues. A simple tutorial can be found out in here.