Question: Was Jesus influenced by Greek thought of the first century? Is it possible that he used it in his teaching?
AP: Probably no more than any man mentioned in the Scriptures during his own day. All of Palestine was more Hellenized than it seemed. Please consult, if possible, my work Biblia y helenismo. La influencia del pensamiento griego en la formación del cristianismo (tr. "Bible and Hellenism: The Influence of Greek Thought in the Formation of Christianity"). I believe there is an electronic version available. It's published by El Almendro (Córdoba 2006).
The thought of Jesus is made up of Hellenistic Phariseeism and its doctrines: Its concept of God and the world beyond the grave are influenced by Greek thought. But Jesus taught them as if they were exclusively Jewish.
TWH: The influence of Greek thought is probably most clearly seen in the writings of Paul and John. For Paul, think about 1 Corinthians, though there we should remember that he is engaged with the Greek-speaking world in somewhat of a different way than one would encounter in first-century Israel. For John, just think about the Gospel written by him and, among the many examples, references to light/darkness (though there is basis for these in the Hebrew Scriptures as well) and other dichotomies. If these disciples were influenced by Greek thought, it seems more than likely that Jesus was as well. But we need to remember that Jesus looked at the world with Jewish eyes through Greek lenses, not Greek eyes through Jewish lenses.
To answer your second question, I believe that John presents an accurate record of the teachings of Jesus. In other words, Jesus taught what John says Jesus taught. If that is the case, it certainly played a role in his teaching. But as Antonio points out, Jesus wasn't leveraging them in his teaching consciously. He saw the world as he saw it––and he saw the world as it actually was. In other words, nothing incorrect about the world from Greek thought carried into the way Jesus understood the world.