Why is my iPhone 5s black screen apple logo and wont charge

From the Apple Support site:

Try putting your device in recovery mode this way:

Plug the device into the computer and hold down both the sleep/wake and home button together until the iTunes logo appears on the phone.

Sometimes when your device is in a reboot loop this is the best way to put it into recovery mode.

Normal Recovery Mode Steps:

1: Disconnect all cables from your device.

2: Hold down the Sleep/Wake button, then "slide to power off" to turn off your device.

2(alt): If the device is unresponsive you may need to power down the device by holding both the Sleep/Wake and Home button together and then release them immediately when the screen goes black.

3: Ensure that the device is fully powered down

4: Press and hold the Home button and plug the device into your computer. If your device doesn't turn on automatically, turn it on. Don't release the Home button.

5: Continue holding the Home button until you see the Connect to iTunes screen.

6: If iTunes doesn't open automatically, open it. iTunes will alert you that it has detected a device in recovery mode.

7: Click the "Restore" Button. Then click "Restore & Update"