An Avatar teaches in two ways.
Firstly His life itself is a teaching.
This was especially so of RAMA AVATAR.
That was the main purport of RAMA AVATAR.
How to live an exemplary life based on Dharma.
In this context Dharma stands for RIGHTOUSNESS.
He was JUST, CORRECT and RIGHTOUS in all what He did and was in His being.
That was the lesson He came to give.
Thus He was called the MARIYATA PURUSHOTTAMA.
That is, He conducted His life within the boundaries of propriety.
Be you perfect as your Father in heaven is perfect.
He lived this statement found in the Gospels, (MATTHEW 5.48).
While Krishna Avatar gave not only a remarkable teaching on the spiritual ART OF LIVING and attaining Salvation, but also lived the teaching as an example.
He lived His life as a classic KARMA YOGIN accepting the challenges of life which came His way, with a serene smile on His lips.
He gathered together, the ancient mystic yoga-knowledge of ancient India into a spiritual discipline for man’s Salvation on earth.
In this wisdom-knowledge, He incorporated the three strands of YOGA, SANKHYA and the UPANISHADIC disciplines into one synthetic system and presented it to mankind as a practical WAY towards Enlightenment.
Furthermore, He also incorporated the spiritual science of KARMA YOGA into the body of His teachings in the Bhagavad Gita.
This science of Karma Yoga was absent prior to the teachings of the Bhagavad Gita.
Hence this was a special and a unique contribution of Lord Krishna.
Prior to this, pagan ritualistic practices and rituals advocated in the Vedas was in vogue.
Hence, He condemned in the Gita, (Chapter 2 verse 42-44) VERY EXPLICITLY the practices and the practitioners of Vedic rituals, as the ignorant blind leading the blind.
He condemns this unequivocally and instead introduces the concept of the daily performance of actions as a ritual and to be done as a Yoga of action or Karma Yoga, as a total discipline for Enlightenment.
It should be noted that a wide spectrum of spiritual knowledge is contained in the Bhagavad Gita.
The genius and the wisdom of Krishna is seen in the Gita.
Such a wisdom-knowledge could not have been given by an ordinary mortal.
Hence solely based on it, I would say that Krishna was an AVATAR, come to give a message to mankind.