From the first game - "ARES! There is nothing you can put in my way to stop me! I will save my family!" Even Kratos' detractors can't help but think damn straight and cheer for him when he shouts this in defiance at Ares after the God of War summons a bunch of Doppelgangers of Kratos himself who are closing in on avatars of his family. Especially since this line comes right after Calliope cries "Daddy!"Heroic Resolve if I've ever seen it. Also counts as a Crowning Moment Of Awesome. Subverted, in that Kratos - despite his best efforts - fails in preventing his family dying a second time, which turns this sorta into a Tear Jerker.
From Ghost of Sparta - Kratos regains his brother Deimos' trust and the two fight side by side as Spartans for the first and last time.
Deimos: A true Spartan never lets his back hit the ground. Right, brother?
Kratos: I will not lose you again. Come! The fight is not over!
"My Lord? I knew you could not be killed! I never lost faith." Leave it to the Last Spartan to inspire humanity in Kratos and Video Game Caring Potential.
The way Kratos rages after Deimos dies and when Zeus insults Pandora, questioning her humanity. When you see Kratos, of all people, literally drip with rage, you know he gives a damn.
A subtle one, but in the E3 demo for Ascension, Kratos is on the boardwalk. Three soldiers/sailors are running towards him, away from spears. Two of them get killed, but as the third spear approaches the survivor, Kratos pushes/shoves him out of the way, saving his life, without even looking. Sure, Kratos rushes into combat afterwards like nothing happened, but in later games, he would've just let the guy get killed. This really shows that, even after just killing his wife and child, Kratos still has some humanity left in him to care for some of his fellow man.
In God Of War Ascension, the Furies try to tempt Kratos back into the service of Ares with a series of increasingly blissful illusions. First, they try to tempt him with a (fake) sex mini game, but Kratos doesnít buy it. Second, they try to appeal to his pride and loyalty to Sparta, something Kratos has always valued in spite of his brutality. It ALMOST works. The real kicker though is the third and final attempt, when he is granted a vision of what his life would be if he had never killed his family. Thus far, every attempt to fool Kratos has set him off like a powder keg. This is the one time heís tempted to accept a life of slavery to the god who ruined his life, even after he discovers that itís all just an illusion. Just being able to dream of better times means that much to him.
Orkos. In a series of black vs. black morality, where the decent or innocent people can be counted with one hand, he is probably the most selflessly motivated character to ever cross paths with Kratos. : Even Athena turned out to be just another power hungry jerk, but Orkos? He remains faithful to his ideals even to the point where it kills him, and he faces even that with dignity. A being born of violence and insanity, he was truly the antithesis to both.
At the end of the game, Kratos makes a pyre for Orkos out of remains of his old home. It's one of the few times Kratos has ever bothered to give someone a proper burial, and in the one place he remembered fondly no less.
The true heartwarming moment in Chains of Olympus when Kratos is reunited with his daughter Calliope. We see Kratos smiles genuinely and happily for the first time in entire series. So heartwarming, seeing how vicious Kratos is when he is in action.
Kratos holding the Oracle's hand as she dies in Ascension. What's more, having seen her death coming, she doesn't even blame him and even forgives him since it more of a Mercy Kill to her having lived so long. She even tell Kratos what he wants to know and after she expires, Kratos actually shows a bit of sympathy for her.