I did this with the Ironclad in a run where I focused on keeping my deck size small and on cards that affected my strength. These were the key cards for me (edit: and J.A.X.)
Limit break was the most important as it loses Exhaust once upgraded and could then be used multiple times per combat.
Demon Form, Spot Weakness, and Inflame all add one additional Strength when upgraded. Flex adds two additional. Demon Form was cumbersome along the way as you generally need to waste an entire turn to cast it.
There are a few relics that help with Strength. I was able to get Brimstone, which adds 2 Strength to you and 1 Strength to all enemies at the start of your turn.
And be sure to snag Strength and Flex potions along the way to get you over the top.