Kratos was born as a Spartan. He once had a family, but slaughtered them by Ares trick, now knowing he has the ashes of them. Kratos then wanted to slaughter the gods.
Kratos's moves aren't mainly from God of War. They're just random moves.
Neutral Special: Blade Slash
Kratos will slash with his blade.
Down Special: Rage
Kratos will use his Blades of Chaos to dash forward.