Ancient Norse and/or Greek
Kratos understands and IS understood by others, implying he speaks ancient Norse, which is later confirmed through dialog between himself and Atreus. In the same conversation we find that he can not at all write it, as he hasn't bothered to learn.
In my head canon, however, whenever he responds to his own hallucination, whether it be Athena or Zeus, he is speaking and hearing Greek. Mimir is well traveled and Atreus has a godly power and understanding of language, so I assume they understand what's being said at these times.
Edit: Also in my head canon, I think whenever Kratos is talking only to himself, such as when he's limping away after his first battle with Baldur, he is speaking in Greek because it's more familiar to him and he has no need to translate his thoughts for others during those times.