Thank you for using Apple Support Communities.
I understand that you are getting a black screen when you try to mirror your iPhones screen through your Apple TV. I would recommend starting with the following support article to make sure that you aren't missing something critical in the setup of the process.
How to AirPlay video and mirror your device's screen to Apple TV
Mirror your iPhone, iPad, or iPod touch
Use Screen Mirroring to see the entire screen of your iOS device on your Apple TV.
- On your iOS device, open Control Center and tap Screen Mirroring.
- Select your Apple TV from the list. If you don’t see your Apple TV, make sure that your iOS device is connected to the same Wi-Fi network as your Apple TV. Need help?
- If an AirPlay passcode appears on your TV screen, enter the passcode on your iOS device.
Your TV uses your iOS device's screen orientation and aspect ratio. To fill your TV screen with the mirrored iOS device screen, change your TV's aspect ratio or zoom settings.
To stop mirroring your iOS device, open Control Center, tap Screen Mirroring, then tap Stop Mirroring. Or press the Menu button on your Apple TV remote.
If you still have the same issue, then you will want to take a look at this article for troubleshooting:
If you can't use AirPlay or mirror your device's screen