In Psalm 22: the picture David gives here is very clear. It is actually without hidden things, but full clear. Some people get saved by hearing Messianic prophesy.
Jesus when on the cross was still in understanding that he was God. He was full of wisdom and no one was able to question that. He quotes scripture on the cross to point to the prophesy. David was a prophet and so he is telling that which Jesus is saying here is what the Psalm meant.
To say God forsook me by Jesus is utter nonsense.
He is putting out a picture of the cry of mankind. For in ch.22 it says God heard him. He is the one who heard the father at his baptism, this is my beloved son in whom I am well pleased.
22:2 shows us it is David crying in the day time. It is through the cry of a person many times that the Holy Spirit will inter mingle his words that are prophetic. After you read the chap.22; see the last v.31.
he hath done this is another way of saying, it is finished.
