Shadow of Fate skeletons are invulnerable to any damage, except briefly after they are exposed to their corresponding lantern color light. You can collect the colors, using your lantern on the Ferry of the Damned by dying in various ways.
GREEN - killed by a skeleton
BLUE - killed by a shark or Megalodon
PURPLE - killed by a snake or Kraken ink
RED - killed by volcano or fire (can be done with firebombs)
PINK - killed by other player
WHITE - killed by lightning
Two ways to get this achievement:
One, pick up 'Skull Seeker' voyages from Duke at any outpost, the map will direct you to an island to kill a skeleton captain. The captain will drop a skull and a letter upon his death, which is an x-marks-the-spot map to another island. Dig up a ritual skull there, and one or more 'Shadow of Fate' skeletons will spawn. The Shadow of Fate skeletons' color is random, and you need to kill one of each color. The free voyages are time-limited, after this event ends the voyages cost 30 doubloons.
Second method: do the Fort of the Damned. See the solutions for the other achievement:
During this event, around 10-12 waves of Shadow of Fate skeletons will spawn and will pretty much guarantee you get all 6 colors.
Updated Jan 28th, 2020: as noted in the comments by Firestormental
, there is currently a bug in the game that prevents the killed shadow skeletons counting towards the achievement after the first 4 waves. As soon as the colours start mixing, the skeletons killed stop being tracked.