This Guide will demonstrate the way to get the Auriel's Bow achievement.
In order to get the Auriel's Bow achievement you have to complete "Touching the Sky" first.
Get the Achievement
After you receive the bow from the quest "Touching the Sky", you actually receive 12 Sun-hungry Elf Arrows and 12 normal Elf Arrows.
The trick is to equip the sun-healed arrows and Auriel's bow and shoot at the sun. I myself left the caves, and as soon as I got out, I equipped both of them and shot them in the sun. You get an achievement and a fair amount of firecrackers after you do it, and it takes about 3 seconds for the special ability to trigger. After that, you will receive your achievement almost immediately.