I am familiar with using the Display Calibration Assistant. I have used it on other external monitors before. Connected to my living room TV via AirPlay using Apple TV. The colors are all off. When I open the displays menu, I am able to open the calibration assistant for the built in display, but not my external Apple TV display. I can see other profiles for other external monitors, but I cannot select them.
I use the THX tune up app on iOS to calibrate my AppleTV over airplay. I've not found a way to get OS X to calibrate things further, but to my eyes, this workaround has been good enough with default settings on OS X.
I use the Calibr8 app for tvOS, which comes with guided instructions and HD.709 professional-grade test patterns.
NB It's not possible to apply colour profile to the device, so you'll need to tweak the settings on your TV to apply any changes.
I like HD Test Patterns for its simplicity and good instructions.