Actually Kratos is not the god of war, Ares is the god of war the game creators just made an imaginary character who was Kratos.

In God of War, Kratos wishes for revenge for his family.

The God of War was Ares. But in the Video Games, you are Kratos, a Spartan Leader, and you have to kill Ares. Once you do that Kratos is the new God Of War. Kratos is the champion of the Gods. That means Kratos fought his gory battles for the Gods and for the glory of Sparta.

kratos

Kratos defeated Ares with the strict goal of vengeance. Then the goddess Athena offered him god-hood which Kratos accepted in turn becoming the God of War.