A Challenging Look

These will not be obtainable once patch 8.0 hits. Leaving this here for posterity.

To attempt to get your new skin you must :

-Be level 110
-Have 35 Artifact Traits
-Complete the new Empowered Trait questline to access the new traits. (Everyone will have the new traits unlocked now that 7.3 is out, but people are reporting the questline must still be done.)
-Have the Mage Tower constructed in your region (it'll be destroyed after 3 days) For a limited time before the launch of Battle for Azeroth the Mage Tower is ALWAYS up!

The first attempt each day is free, subsequent attempts require 100 Nethershards.
Update: you get one free try each time day the building is up.

Speak with War Councilor Victoria to start the quest. She's right outside the Mage Tower.

Does anyone know if you have to wait for all bosses in violet hold to spawn in order to complete the 4th color?

Original unlock is still unknown; however, the requirements for the following 3 appearances are displayed on the PTR:

Tint 2: Defeat Heroic Kil'Jaeden using this Challenge Artifact Appearance.
Tint 3: Win 10 Rated Battlegrounds using this Challenge Artifact Appearance.
Tint 4: Defeat all Legion Dungeon Bosses using this Challenge Artifact Appearance.