Here is the full list of all 13 The King's Bird achievements.
Restore the Barrier and usher in a new era of peace
Found all of the spirit birds across the world
Found the elusive purple tree
Destroy the Cage and usher in a new era of freedom
Win the Battle for the Last Kingdom
Complete all levels in the Sky Kingdom
Found all the spirit birds in the Sky Kingdom
Complete the Fallen Kingdom
Escape the Fallen Kingdom in under 10 minutes
Complete all levels in the Lake Kingdom
Found all the spirit birds in the Lake Kingdom
Found all the spirit birds in the Forest Kingdom
Complete all levels in the Forest Kingdom