Q: Is Krishna the Supreme Personality of Godhead or just a teacher like Jesus?
Krishna is not Parabrahman, the Absolute Being, and Jesus is not the Godhead, either. Nor are they “just teachers,” either.
Both Krishna and Jesus were perfect siddhas, thoroughly deified persons, truly god, having evolved through all the worlds of relative existence and transcended them in total union with the Absolute. Though finite, their consciousness and power is infinite. And this is true of all incarnations of God (avatars) such as Rama and Buddha. (See the book Robe of Light.)
A totally liberated being, a siddha, is far beyond the concept of God held by human beings, for being finite humans cannot at all even begin to correctly conceive of God. This is why the truly God-realized never attempt to describe or explain the nature of God.
Words from a Master
As Sri Ramakrishna said,
“What Brahman is cannot be described. All things in the world–the Vedas, the Puranas, the Tantras, the six systems of philosophy–have been defiled, like food that has been touched by the tongue, for they have been read or uttered by the tongue. Only one thing has not been defiled in this way, and that is Brahman. No one has ever been able to say what Brahman is.
“A man had two sons. The father sent them to a preceptor to learn the Knowledge of Brahman. After a few years they returned from their preceptor’s house and bowed low before their father. Wanting to measure the depth of their knowledge of Brahman, he first questioned the older of the two boys. ‘My child,’ he said, ‘You have studied all the scriptures. Now tell me, what is the nature of Brahman?’ The boy began to explain Brahman by reciting various texts from the Vedas. The father did not say anything. Then he asked the younger son the same question. But the boy remained silent and stood with eyes cast down. No word escaped his lips. The father was pleased and said to him: ‘My child, you have understood a little of Brahman. What It is cannot be expressed in words.’