In order to unlock this you will need to obtain a series of tokens only available from playing the Warlord fate. This achievement is obtainable even for those who have not purchased the DLC.
I recommend playing Jack of Skulls for these tokens with the exception of Warlords 2 and 3 (which require more cards to help you obtain the necessary items before finding the encounter). This affords a good chance of finding them without having to slog through the more challenging levels later on.
Questchain
Warlord 1: No requirements, merely land on this encounter for the token.
Warning Signs: Select "A) Attempt to cross the valley floor." You will then need to pick a card 3 times. It does not matter whether you pick Success, Failure or Huge Failure* for the purposes of acquiring this token. Once you have survived this you will receive the token.
*Failure will incur 3 Health Pain cards and Huge Failure will incur 6 (Success causes nothing to happen) so approach this with reasonable health if you can to avoid dying.
Warlord 2: You will need the Disciple's Ring in your inventory when you find this encounter to obtain the token.
Best way to increase your chance of this happening is to remove all other rings from your deck and place in encounters such as "The Lovers" (demand a reward and they will nearly always give you rings) as well as other equipment-orientated cards such as Treasure Chest/Cursed Treasure. I played the King of Dust level for this one.
Captain's Woes: Select "Yes, help him get his crew back". You will then need to defeat 1 Lich and 1 random skeleton card.
The Curse of Davey Jones: Pick a success card (2/4).
(Huge) failure is not an option! Dashboard quit if you fail and reload, otherwise it is instant death.
Warlord 3: You will need the full Skeleton King set (Skeleton King Sword, Skeleton King Shield and the Skeleton King Helm) in your inventory upon reaching this encounter, after which the achievement will be yours.
The most effective method for acquiring the set is to remove all other weapons, shields and helms from your deck and refill the deck with additional rings and artefacts. This way certain encounter cards guarantee you a skeleton armour piece. For example Twisted Canyon will guarantee you the Skeleton Sword.
You can also leave and re-enter shops (or if low on food reload the game) to refresh the stock and allow you more chances at purchasing any items you're missing. Credit to DarkLordRaptor for reminding me to include this.
I also obtained this particular token on King of Plague, although most of the later bosses should give you sufficient extra cards. I would not recommend Endless as it could get very frustrating.
At this point you can also obtain the King of the Undead achievement from the main game's achievement list if you make sure the items are equipped
Additional Notes:
You can forfeit as soon as you have the token if the above tokens are the only ones you are going for in a given run. Also, if necessary, you can always exploit the dashboard quit at any time which will, upon reloading the game, return you to the point before your last encounter.
If there is anything I have missed or hints I could include, please let me know in the comments and I will edit the solution.