Krishna is an historical person who appeared on this earth in India 5,000 years ago. He stayed on this earth for 125 years and played exactly like a human being, but His activities were unparalleled.
Sri Krishna is the Supreme Personality of Godhead.
Because He conforms in exact detail to descriptions of the Supreme Being, the Godhead. In other words, Krishna is the Godhead because He is all-attractive. Outside the principle of all-attraction, there is no meaning to the word Godhead. How is it one can be all-attractive? First of all, if one is very wealthy, if he has great riches, he becomes attractive to the people in general. Similarly, if someone is very powerful, he also becomes attractive, and if someone is very famous, he also becomes attractive, and if someone is very beautiful or wise or unattached to all kinds of possessions, he also becomes attractive. So from practical experience we can observe that one is attractive due to 1) wealth, 2) power, 3) fame, 4) beauty, 5) wisdom, and 6) renunciation. One who is in possession of all six of these opulences at the same time, who possesses them to an unlimited degree, is understood to be the Supreme Personality of Godhead. These opulences of the Godhead are delineated by Parasara Muni, a great Vedic authority. Finally, all of these divine attributes reside eternally within the spiritual body of Krishna.