To unlock this you will need to obtain the 3 tokens available from playing the Curse of the Lion Prince fate. This fate starts you off with only 40HP, but each enemy you kill will increase your max HP by 2 (regardless of enemy type).
Note: All of the tokens require you to increase your maximum HP to a certain level in order to receive the token. Unfortunately, and despite a great deal of digging, I can find no concrete evidence of what exactly the requirement is - although it is certainly over 100. For each token I will include the lowest HP that I know will work as well as the other requirements. If readers could comment as to how much they had for each token that would be very helpful.
Credit to Fear Catalyst who points out in the comments that it is possible to exploit encounters such as the Kraken by repeatedly killing the minions spawned during the fight. Another exploit to boost your HP is to repeat ambushes by revisiting shops.
For all of these tokens I played King of Dust.
Test of Pride 1: This token simply requires you to have a high enough maximum HP. This can be obtained with 125hp, credit to I Argonaut I. It has also been confirmed that 117hp is too low, credit to the rather unlucky Phr4nk0 for this.
Test of Pride 2: For this token you will need to have both a high maximum and
current HP. Therefore make sure you are fully healed when you come to this card. It appears that 129hp is enough for this, credit to I Argonaut I.
Test of Pride 3: As well as having a high maximum and current HP, you will also need to be free of curses. Make sure to pick a level where you only begin with one curse (if you are not playing King of Dust) and ensure that the Soul Gem is included in your deck. This will allow you to pay 5 food to remove your starting curse. It may be necessary to modify your deck to reduce the chances of being cursed (or use the exploit trick if you can avoid the curse by playing differently!*).
It has been confirmed that 140hp is enough for this. Credit to UpgradePolecat.
For these tokens I filled my deck with a large number of encounter cards (just to be on the safe side) but also cards that often give increases to maximum HP such as The Maiden/Ghost of the Sea and even Secret Society if you're feeling lucky. In particular the Soul Gem which is essential to the third token but also useful for the previous 2. Narcotic Radar has pointed out in the comments that it may be worth removing all rings except Ring of Experience/Defence/Fulfilment/Poverty and Saint's Ring - if you find yourself struggling these rings should be helpful.
Unlike many other encounters it is possible to return to these tokens afterwards if you failed to reach the requisite health. This allows you to continue exploring your current floor in the hopes of reaching the necessary amount. The tokens will give you hints as to how far from the required max HP you are if you read the text carefully. For example, in Test of Pride 1, if you have not achieved the required max HP you will receive one of two responses (credit to the Hand of Fate wiki):
(low max health) "Woe be upon you. You are not yet fit for our pride. When we next meet it would benefit you to have hunted more of the prey that infest this land."
(insufficient max health) "You make progress towards becoming a Pride member but you still lack strength. Hunt more prey and you will be ready soon."
*Also remember: 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.