Step 1. Complete the artifact questlines for the specs you intend on playing. I highly recommend finishing both fire and frost, as going back to do them after this will remove your cool new arcane form.
Step 2. Start the questline for Aluneth - Item - World of Warcraft
Step 3. Complete the quest Harnessing the Arcane - Quest - World of Warcraft (part of the artifact questline, obviously)
Step 4. Once you have the quest Arcane Unleashed - Quest - World of Warcraft in your quest log, teleport out of Coldarra (I used Dalaran - Broken Isles, but anywhere should work)
Step 5. Enjoy your new appearance.