In order to fully upgrade Dante's health, you'll need to complete several challenge rooms. Some of these rooms reward you with a Health cross piece, others with a Devil trigger cross piece. Once you gather four of either, a bit of Dante's stats will be increased.
However, to full upgrade your health, you'll need to buy all the pieces from the store as well. They start out cheap, but will increase in price once you've purchased one.
This will help towards the "Absolutely crazy about it" achievement.