Shakespeare's depictions of historical individuals draw from sixteenth-century chronicles, where the depiction of Richard III as physically monstrous already had plenty of currency. Holinshed's Chronicles of England (~1587) purported to trace English history from the time of William the Conqueror, and were relatively current by the time Shakespeare was writing -- these are one of the most direct sources for Shakespeare's plays. Shakespeare drew elements for King Lear, Macbeth, and Cymbeline from this same text as well as other English history plays like his depictions of Richard II and Richard III. From Holinshed you can trace the visible skeleton of Shakespeare's narrative for Richard of Gloucester's life -- his political treachery and clearing-away of obstacles to the throne via murder, the murder of his nephews, his attempt to marry his niece, the battle of Bosworth Field -- and the assertion that Richard III was physically unappealing as well as morally unappealing:

As he was small and little of stature, so was he of body greatly deformed; the one shoulder higher than the other; his face was small, but his countenance cruel, and such, that at the first aspect a man would judge it to savour and smell of malice, fraud, and deceit. When he stood musing, he would bite and chew busily his nether lip; as who said, that his fierce nature in his cruel body always chafed, stirred, and was ever unquiet: beside that, the dagger which he wore, he would (when he studied) with his hand pluck up & down in the sheath to the midst, never drawing it fully out: he was of a ready, pregnant, and quick wit, wily to feign, and apt to dissemble: he had a proud mind, and an arrogant stomach, the which accompanied him even to his death, rather choosing to suffer the same by dint of sword, than being forsaken and left helpless of his unfaithful companions, to preserve by coward lie flight such a frail and uncertain life, which by malice, sickness, or condign punishment was like shortly to come to confusion.

Holinshed derives his account largely from earlier sources, including from the historical writing of Thomas More. Thomas More's incomplete History of King Richard III (~1513-1522) is really where the juice starts flowing in depictions of Richard III as monstrously deformed. More reiterates assertions about Richard's murder of his nephews, and adds the drowning of George Duke of Clarence in a butt of Malmesey wine. His Richard is a cold-blooded and martial character, not only crooked-backed and hard-favored but a breech birth born (in More's rather coy phase) "not untoothed".

Richard, the third son, of whom we now treat, was in wit and courage equal with either of them, in body and prowess far under them both: little of stature, ill featured of limbs, crooked-backed, his left shoulder much higher than his right, hard-favored in appearance, and such as is in the case of lords called warlike, in other men called otherwise. He was malicious, wrathful, envious, and from before his birth, ever perverse. It is for truth reported that the Duchess his mother had so much ado in her travail to birth him that she could not be delivered of him uncut, and he came into the world with the feet forward, as men be borne outward, and (as the story runs) also not untoothed. [...] He was close and secret, a deep dissembler, lowly of countenance, arrogant of heart, outwardly friendly where he inwardly hated, not omitting to kiss whom he thought to kill; pitiless and cruel, not for evil will always, but for ambition, and either for the surety or increase of his estate. Friend and foe was much the same; where his advantage grew, he spared no man death whose life withstood his purpose. He slew with his own hands King Henry the Sixth, being prisoner in the Tower, as men constantly say, and that without commandment or knowledge of the King, who would, undoubtedly, if he had intended such a thing, have appointed that butcherly office to some other than his own born brother.

Polydore Vergil's Anglica Historia (~1534-1570) likewise gives a depiction of Richard III's life and reign that would have been accessible to Holinshed and perhaps Shakespeare. Vergil gives a vivid and detailed account of Richard's machinations against Elizabeth Woodville and particularly against his nephews, as well as intimating that Richard murdered his first wife Anne in order to marry his niece. Holinshed's description of Richard's mannerisms is drawn pretty much directly from Anglica Historia:

He was slight of stature, misshapen of body, with one shoulder higher than the other, and had a pinched and truculent face which seemed to smack of deceit and guile. While he was plunged in thought, he would constantly chew his lower lip, as if the savage nature in that miniature body was raging against itself. Likewise with his right hand he was constantly pulling the dagger he always wore halfway in and out. He had a sharp, clever, wily wit, fit for pretence and dissimulation. His spirit was lively and fierce, and did not fail him even in death. For when abandoned by his own men, he preferred to take up the steel than to save his life by shameful flight, unsure whether he might soon lose it by disease or by suffering his comeuppance.

Vergil also gives Richard III an unusual-looking arm, but it's only remarked on in the context of Richard revealing it before a group of men as a support for his allegations of malicious witchcraft against Elizabeth Woodville. A number of other 16th century chronicles iterate a similar set of points that would have been in literary circulation at the time Shakespeare wrote his history plays -- Richard of Gloucester was treacherous and warlike, Richard of Gloucester was maliciously clever, Richard of Gloucester was a small man with uneven shoulders who murdered his nephews among others. Edward Hall describes the smothering of the princes in the Tower and their secret burial.

So there was nothing novel to a reasonably well-lettered Elizabethan audience member about depicting Richard III as either wicked or hunchbacked, and another hypothetical history play composed in the same vein as Shakespeare's would likely have depicted a similar figure in those two respects. In his play Richard III Shakespeare isn't simply uncritically replicating previous descriptions of Richard III's dispositions and mannerisms, lovingly reproducing so-called Tudor propaganda, but nor is he originating certain elements out of whole cloth; he trims some elements that might be difficult to convey on stage and elaborates on others as it pleases him. He is drawing from accepted historical accounts and tailoring them to the dramatic requirements of Early Modern drama. None of these accounts emphasize Richard's deformity to the degree that Shakespeare's play does; Shakespeare situates Richard's deformity among his motivations for malice, but he also gives him a little psychological backstory beyond "he's ugly inside, so he's ugly outside". This might be the distinguishing factor that sets Shakespeare's villainous Richard III from the Richard III of various accepted chronicles. Richard's deformity sets him at odds with the world around him from boyhood onward, making villainy seem like an appealing means by which to make up for his deficiencies, rather than merely serving as a badge of innate freakishness or being a plain unvarnished historical fact with no bearing on Richard's morals or personality.

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