Kratos is seen by many people in forums as a one dimensional character only seeking vengeance which has actually spread through media. Many people just go with it despite not having played all the games, going by just a few games they have played or some footage they saw of Kratos ripping the head off of a gargoyle. That's not true, as he is tricked by the gods, lied to and used as a pawn to further their motives. There are spoilers ahead.
See some of the things I will say and you may be able to know his motives a bit better. He was tricked by Ares into killing his own wife and child and then the ashes of his wife and child were stuck to him by a witch who cursed him into having to live with it for the rest of his life and given the title, the Ghost of Sparta. That is brutal.
He then seeks to break his bond with Ares by serving the Olympian Gods and does so for ten years, killing huge creatures and taking the lives of those that threaten Greece yet the Gods don't free him from his punishment. He saves Helios and brings back the sun but gets nothing. He is reunited with his daughter by Persephone but in return he has to allow the world to be destroyed so he reluctantly leaves his daughter again to save the world. He still gets nothing from the Gods.
Kratos is then sent on a quest to kill Ares who plots to overthrow Olympus and the Gods promise to forgive him for his sins which was perpetuated by Ares and not his fault. He does as told but then they back out of it saying that he is forgiven for his sins but has to live with their memories.
After all this he becomes the God of War and then has flashbacks about his mother living in Atlantis. He goes there and is attacked by Scylla, a servant of Poseidon, an Olympian God. See they just don't let him live in peace. Then when he does meet his mother she turns into a hideous beast and he has to kill her too. She tells him about her brother who was taken away as a child by Thanatos, the God of Death, and he goes to find him. He finds Deimos who is angry with him and attacks him. After a short fight Thanatos takes Deimos away and nearly kills him but is saved by Kratos just in time. The two brothers then go on to fight Thanatos who kills his brother.
Kratos wins and is angrier than ever before. He goes on a rampage which is understandable as all the Gods treat him terribly and kill his family. He goes to Rhodes and is attacked by the Collosus which comes alive. Zeus sends him the Blade of Olympus to destroy the Collosus as if he was trying to help him. To use the sword he had to give all his power to it and become mortal again. He destroys the Collosus but after escaping as the Collosus dies one of his hands falls on Kratos severely injuring him. He goes to the sword to get back his godly power but Zeus arrives and takes it, revealing that he was the one who bestowed life to the statue and it was all a trick in order to turn Kratos into a mortal again. He then shoves it into Kratos torso when he is severely wounded. After all this Kratos seeks vengeance on Zeus. The other Olympian Gods weren't his targets but they stood between him and Zeus which is why he had to kill them. Ares, Persephone, Thanatos and Zeus were the only true enemies of Kratos, the rest just got in his way.