set up closed captions on streaming

On iPhone? (You selected that as relevant device for this question.)

1. Set while viewing from the video playback interface for subtitles and closed captions, or
2. Set a default for any future video in Settings﹥Accessibility﹥Subtitles & Captioning﹥Closed Captions + SDH﹥

Note that not all content supports CC, as it has to be included in the stream.

Note that AirPlay may not use the settings of the sending device, but instead the settings of the receiving device.

Typically each app on the Apple TV is a little different for starting closed captions. However, most have some type of option in the far right bottom corner once you start a show. If you are able to view the progress bar, that's when the close caption box appears in the bottom right of your program.

We have no idea if you are referring to the Apple TV app or a physical Apple TV. Please post a complete question.