As a child, Krishna kept his divinity a secret, so that Kamsa wouldn't know that Vishnu had been born on Earth to kill him. When Krishna was born, he had four arms and all the ornaments of Vishnu, but Devaki asked him to appear like an ordinary human child so Kamsa wouldn't know his true identity, as described in the Srimad Bhagavatam:
Your form as Viṣṇu, the Supreme Personality of Godhead, is appreciated by yogīs in meditation. Please make this form invisible to those who see with material eyes. O Madhusūdana, because of Your appearance, I am becoming more and more anxious in fear of Kaṁsa. Therefore, please arrange for that sinful Kaṁsa to be unable to understand that You have taken birth from my womb. O my Lord, You are the all-pervading Supreme Personality of Godhead, and Your transcendental four-armed form, holding conchshell, disc, club and lotus, is unnatural for this world. Please withdraw this form.
So Krishna took various measures in his childhood to hide his divinity, for instance making his foster mother Yashoda forget about seeing the Universe in Krishna's mouth, as described later on in the Srimad Bhagavatam:
Mother Yaśodā, by the grace of the Lord, could understand the real truth. But then again, the supreme master, by the influence of the internal potency, Yoga-māyā, inspired her to become absorbed in intense maternal affection for her son. Immediately forgetting Yoga-māyā’s illusion that Kṛṣṇa had shown the universal form within His mouth, mother Yaśodā took her son on her lap as before, feeling increased affection in her heart for her transcendental child.
And a later chapter of the Srimad Bhagavatam describes how Krishna told his friends to close their eyes before he swallowed up a forest fire.
It's only after Krishna went back to Mathura and killed Kamsa that people started to learn that Krishna was an incarnation (avatara) of Vishnu, mainly through the words of famous sages who openly declared this fact to the world; in the Bhagavad Gita, Arjuna mentions four sages who have proclaimed Krishna's divinity:
You are the Supreme Personality of Godhead, the ultimate abode, the purest, the Absolute Truth. You are the eternal, transcendental, original person, the unborn, the greatest. All the great sages such as Nārada, Asita, Devala and Vyāsa confirm this truth about You, and now You Yourself are declaring it to me.
As a result, a great many people accepted Krishna's divinity during his lifetime. For instance, in the Sabha Parva of the Mahabharata, Bhishma tells Yudishthira to make Krishna the chief guest of his Rajasuya Yagna, giving this as his justification:
He that approveth not the worship offered unto Krishna, the oldest one in the universe, deserveth neither soft words nor conciliation.... This one (meaning Krishna) here, of undefiled glory, deserveth to be worshipped not by ourselves alone, but being of mighty arms, he deserveth to be worshipped by the three worlds also.... The whole universe without limit is established in him of the Vrishni race. Therefore do we worship Krishna amongst the best and the oldest, and not others.... Krishna is the origin of the universe and that in which the universe is to dissolve. Indeed, this universe of mobile and immobile creatures hath sprung into existence from Krishna only. He is the unmanifest primal cause (Avyakta Prakriti), the creator, the eternal, and beyond the ken of all creatures. Therefore doth he of unfading glory deserve highest worship. The intellect, the seat of sensibility, the five elements, air, heat, water, ether, earth, and the four species of beings (oviparous, viviparous, born of filthy damp and vegetal) are all established in Krishna. The sun, the moon, the constellations, the planets, all the principal directions, the intermediate directions, are all established in Krishna.
And since you asked about Draupadi, her plea to Krishna acknowledges the fact that he is Vishnu:
O Govinda, O thou who dwellest in Dwaraka, O Krishna, O thou who art fond of cow-herdesses (of Vrindavana). O Kesava, seest thou not that the Kauravas are humiliating me. O Lord, O husband of Lakshmi, O Lord of Vraja (Vrindavana), O destroyer of all afflictions, O Janarddana, rescue me who am sinking in the Kaurava Ocean. O Krishna, O Krishna, O thou great yogin, thou soul of the universe, Thou creator of all things, O Govinda, save me who am distressed,--who am losing my senses in the midst of the Kurus.
And in the Udyoga Parva of the Mahabharata, when Krishna goes to Duryodhana's court as an envoy of the Pandavas, the Kauravas remain silent after Krishna's offer of peace, so Vishnu's incarnation Parashurama (who happens to be in the court at the time) admonishes them, telling them that Krishna is a reincarnation of Vishnu's incarnation sage Narayana, as I discuss in this answer. Sage Kanva makes a similar statement to Duryodhana.
And to top it all off, Krishna revealed his divine Vishwarupa form to everyone in the court before he left:
And as the high-souled Sauri laughed, from his body, that resembled a blazing fire, issued myriads of gods, each of lightning effulgence, and not bigger than the thumb. And on his forehead appeared Brahman, and on his breast Rudra. And on his arms appeared the regents of the world, and from his mouth issued Agni, the Adityas, the Sadhyas, the Vasus, the Aswins, the Marutas, with Indra, and the Viswedevas. And myriads of Yakshas, and the Gandharvas, and Rakshasas also, of the same measure and form, issued thence.... And on his diverse arms were seen the conch, the discus, the mace, the bow called Saranga, the plough, the javelin, the Nandaka, and every other weapon, all shining with effulgence, and upraised for striking.... And beholding that awful form of the high-souled Kesava, all the kings closed their eyes with affrighted hearts, except Drona, and Bhishma, and Vidura, endued with great intelligence, greatly blessed Sanjaya, and the Rishis, possessed of wealth of asceticism, for the divine Janardana gave unto them this divine sight on the occasion.
It's amazing that despite all of this, Duryodhana still refused to accept Krishna's divinity even after seeing it with his own eyes! It's only at some point during the Mahabharata war that Duryodhana finally admits to himself something along the lines of "Perhaps the sages are right, and Krishna really is Vishnu."
