The Lamplighters League

This achievement requires you to first find and unlock Madame Mei as a team member. In her upgrade tree, she not only has the upgrades necessary for unlocking the ability to upgrade Undrawn Hand Cards to level IV and V, but she also has the upgrades for finding more cards, increasing rarity drops, and increase in Ink upon disposal.

To level one card up it requires the following (will update with accurate numbers if someone needs it):

Level I to II: 32 Ink
Level II to III: ~125 Ink
Level III to IV (Once Unlocked): ~450 Ink
Level IV to V (Once Unlocked): 2048 Ink

To unlock level V, you first need 2x Kings Aether (KA) to even purchase. KA can be found as an end of mission reward.

Note: I found Seric Steel once in a map as a random pickup, never found KA, but not sure if KA can also be found. Thanks to BloodyCurse, KA can be found in levels, but are rare.

Also, thanks again to BloodyCurse, cards can be upgraded for free if you draw the same card at end of level.

Hope this helps!