You'll unlock the Claws of Hades weapon after defeating (surprise, surprise) Hades, the second boss in the game, following Poseidon.
Press right on the D-Pad to select the weapon and then R2 to summon whatever soul you have equipped - you can do this at any time, not just during combat, but it uses magic. To equip a soul, go to the Power-Up menu and press triangle to see the souls you have. You'll need to fully upgrade the Claws of Hades to level 5 to unlock all of the souls.
On God (Normal) difficulty with a fully-upgraded magic meter, I could summon seven of the nine souls before running out of magic, so an easy way to get this is to simply summon all of the souls one-at-a-time next to a magic-refill chest.
Here are the following nine souls that you can summon and at what level they are unlocked:
Level 1
Cerberus Mongrel, Olympus Sentry, Olympus Archer
Level 3
Gorgon Serpent, Olympus Fiend, Chimera
Level 5
Cyclops Berserker, Centaur General, Siren Seductress