how do i play my itunes music library through fire tv stick?

if you’ve already enabled Apple Music on an Amazon Echo, then your Fire TV device will automatically be able to use Apple Music. However, if you just now are looking to add the Apple Music, you’ll need to enable the skill. 

1. Go to the apple music skill in a browser or the Alexa app.
2. Select the "Enable" button to enable the skill.
3. Sign in to your Apple Music account with your Apple ID.
4. Allow the Access Request.
5. You can choose Apple Music as the default music provider for Alexa, if desired.
6. Grab your Fire TV remote, tap the mic button, and tell Fire TV to "Play Apple Music".
7. If it’s working correctly, you’ll hear something like: “Playing your song on Apple Music".

APPLE

Apple Music hitting the Fire TV is another sign that Apple is slowly moving away from its old approach of forcing people to own an Apple device to use its services. It was only in December that Apple Music came to Echo devices, and with Apple's video streaming service due to launch, we suspect the company will want to continue opening up its services to other platforms. Best way to fix Amazon Fire Stick remote Not working