Hunter King Coins are a crafting material that can only be obtained through the Arena and the completion of Arena Quests. Arena Quests can be obtained at the Gathering Hub and by talking to Arena Lass, the character in the blue outfit at the Arena Counter.
Arena Lass will provide you with the quest.
There is some disparity as to which monster is the easiest to beat in order to farm the coins, but the consensus seems to shift between Dodogama and Barroth, so take that into account when picking the quest.
The quest for slaying Dodogama.
The quests have timers that determine your performance, but as long as you don’t die, you have a high chance of obtaining Hunter Kings Coins; however, the better you perform, the better your odds will be of obtaining more coins.
Because the quests take place in the Arena, you must choose from a set of weapons that are available to you (i.e., you can’t use your own), so make sure you pick what best suits you.
Here are some other Monster Hunter: World guides you might find useful:
Light Bowgun Build Guide Piercing Claw Guide Multiplayer Expeditions Guide Monster Hunter: World Guide - Teostra’s Weakness Guild Cards Guide