I got to dungeon level 50 and amassed some 2000 gold. On the character select screen, I can level up starting at like 60 gold and +5 gold per level after.
The swordsman gets +1 HP per level up, but there appears to be a max of 99 HP. You can easily collect +1 max HP per level (over 50 dungeon levels, that'll add up). So, I'm thinking it only makes sense to level up so far.
So, finally, is there a level cap?