how to use Apple TV 4K hardware with HDMI connection to TV to stream sound from TV to headphones

I am not sure what exactly you are asking. Are you asking how to route audio from your cable, antenna, and/or satellite TV through the Apple TV? This is not possible. You would need to connect a Bluetooth transmitter to an audio output port on your TV. Only audio originating from Apple TV can be used with Bluetooth headphones on Apple TV.

Hello,

What type of headphones are you intending on using?

Hello,

You can pair Bluetooth headphones in Settings > Devices > Bluetooth.