What message does Cole's statement, "I saw that my righteousness would not save me; then I was convinced of the doctrine of Election and went right to quarrelling with God about it, because all that I could do would not save me" reveal about Whitefield's teachings?

This Cole's statement reveals about Whitefield's teachings of the doctrine of predestination: nothing of what a man could do would save him, because God has chosen since the beginning who to save and who to doom.

This is a characteristic of Calvinism, produced against the practice of buying forgiveness that was extended in Christianism at that time. Only faith that God has chosen one to saved is what could save a man according to this doctrine, not if he acted good or bad.