Narratively, Chains of Olympus actually comes before God of War in the timeline of Kratos universe. In terms of real world game releases, it comes third in our God of War series, and also makes the series debut on the PSP. Though one of the earliest PSP games to hit the market, Chains of Olympus would be one of the top rated and most successful games on the platform, only surpassed in quality by its successor (Ghost of Sparta) and a few other choice picks. The crux of Chains of Olympus is that Kratos is a tool of the gods, particularly Ares himself, and when the Pantheon themselves are threatened, it is up to this mostly mortal man to fix their problems. He does so with his twin Blades of Chaos, and while his killing isnt nearly as rampant and bloody as in any other game in the series, this is technically his heroic debut, so we can forgive him a shorter death list.
In-game chronology would have the Basilisk as Kratos first truly legendary kill. This is the first real boss fight in Chains of Olympus, and also the biggest enemy he fights in the entire game. In Kratos world, the Basilisk is a large, dinosaur-like beast that breathes fire and has no stoning properties. Its as tall and as wide as a building, has a Spinosaurus-like protrusion along its back, and bears the mottled skin of one of the great lizards. It also has large, webbed arms, though it never shows any indication that it can fly. This image is in stark contrast to most of those in Greek mythology, but things are usually bigger and badder in the Kratos-verse, so this is hardly any surprise. In legends, the basilisk ranges from a diminutive snake-like creature to something relatively cow-sized, and has even been represented multiple times as very chicken-like in appearance. However, its rarely, if ever, a behemoth, and in no accounts that I could find did it ever breathe fire. It has had the ability to spit venom, but its singular characteristic has always been its ability to turn victims to stone with a mere gaze. Perhaps Ready at Dawn didnt wish to re-utilize this mechanic after the gorgon fights of God of War I and II.
Regardless of its mythical variance, the basilisk fight, in a style reminiscent of every God of War game, starts Chains of Olympus off with a roar, and somehow Kratos manages to break the great reptiles neck.
As with just about every single God of War title, at some point Kratos is either thrown or forced into the Underworld where he invariably sloshes around slaying things that are supposed to be already dead. As the mythology would tell it, Charon is the ferryman who trucks souls across the River Styx and into Hades proper. Charon is generally depicted as an old, hooded figure, and not necessarily a warrior or magician capable of fighting. However, he must have some power because he is able to deny just about any but the most persistent of mythological figures access to the Underworld. This is what happens to Kratos when he first comes upon his boat, toying with the Ghost of Sparta as though he were a child. Kratos eventually returns and not only beats Charon to death, but rips his mask off revealing the hideous geezer beneath.
There is no mention in later games the effect that not having an underworld ferryman had on the process of souls crossing Styx. Presumably, theres a very long line just waiting to get in.
Persephone might be the most peculiar kill for Kratos, for a few reasons. Its not strange because shes a fairly major god, but because he kills her so early into his god-slaying career. By all rights, this lone mortal should not have stood a chance, walking into the lair of a deity and slaying her with impudence. Nevermind the fact that Hades probably would have been nearby and exacted a swift revenge, a fight that Kratos could not have withstood in his pre-ascension days (though stranger kills have happened).
In traditional mythology, Persephone is the child of Demeter and Zeus (incest!), but is abducted in the famous Rape of Persephone myth by Hades, and made to live in the Underworld for eternity with only occasional visits to Olympus and the world above. At these times, nature flourishes and thus the reason for Spring.
The entire Chains of Olympus story revolves around Persephones revenge, and how she kidnapped Helios, the Sun God, and with her ally Atlas, the Titan holding up the earth, tried to destroy the pillar supporting Olympus and thus doom all the gods. Kratos, as a tool of the gods at this point, kills her for her brash actions. How this affects the coming of Spring and nature in general is never made mention of in any subsequent narratives. As if Kratos cares about some namby-pamby Spring anyway.