Shri Krishna is the Supreme Personality of Godhead. This is affirmed in history and the Veda's, in sanskrit this is known as...

Kṛṣṇa and the Supreme Personality of Godhead are identical. Therefore Lord Kṛṣṇa is referred to as "Bhagavān" throughout the Gītā. Bhagavān is the ultimate in the Absolute Truth. Absolute Truth is realized in three phases of understanding, namely Brahman or the impersonal all-pervasive spirit; Paramātmā, or the localized aspect of the Supreme within the heart of all living entities; and Bhagavān, or the Supreme Personality of Godhead, Lord Kṛṣṇa. In the Śrīmad-Bhāgavatam this conception of the Absolute Truth is explained thus:

vadanti tat tattva-vidas tattvaṁ yaj jñānam advayam

brahmeti paramātmeti bhagavān iti śabdyate.