Each turn, he'll count down to his ultimate move (4, 3, 2, 1, 0/Attack). If you're at max health, it'll just bring you down to 1 HP, and then you can use a party-healing item (Finless Cola should be fine, you don't need to waste your Croaka Cola). If you're at less than full-health, it'll insta-kill you (which can't be protected against), so make sure you top yourself off when he reaches 1.
Just ignore the elemental crystals and focus solely on him. When you kill him, the crystals will restore him at full health, shattering themselves in the process (all of them at once, not one at a time). Don't kill the crystals, because they'll buff Culex and the other Crystals when they die, unless they're shattering themselves to revive him.
After that, just kill him again. He'll change his attack to Final Claw, which does 9999 damage to one of your party members. On the bright side, this can be guarded against, unlike his other attack, so you just need to block it. Or revive whichever party member died.
I recommend using Mario, Bowser, and Peach.
