The quest chain itself is not that long, and takes about 30 minutes to finish. This will lead to you taking the portal to Darkshore where you will have to complete a few quest, following by a short single-player scenario that tells the story of how Tyrande became the Night Warrior. Accept this, then head to Stormwind to talk with Maiev Shadowsong.
There you will be approached by Tyrande's owl Dori'thur with a quest. Start off by logging in to a level 110 character, and head for Bolarus. All Night Elves can use this customization, but only after at least one character has unlocked the account-wide feature. Patch 8.1 added the ability for Night Elf players to change their eyes into the Night Warrior Eyes, giving them dark-looking eyes instead of the regular yellowish eyes they come with.