This is a great question. It took scholarship a long time to untie this knot. That said, this question is not easy to answer nor is it definitive. The fact is, with many things in Late Antiquity, some of the details we’ll just never know with our current knowledge base.

First, you have, in effect, asked four questions: 1) Did Justin rise from poverty? 2) Was he illiterate? 3) How did he achieve such a high position? And, 4) How did the elites of Constantinople react to his accession? I will answer each of these question to the best of my ability in this order. Starting note: Keep in mind that Procopius is pretty hostile of Justin and Justinian. I have written the conclusion of each section into the first few lines, in case anyone does not feel like reading the whole answer.

1) Was he low-borne? Probably, but we just don’t know to what extent he was so. It has, most certainly, been exaggerated though. The notion of Justin having come from poverty is propagated by Procopius’ Secret History (aka, Anecdota).

“[2] When Leon was holding the imperial power in Byzantium, three young farmers, Illyrians by race, Zimarchus, Dityvistus and Justinus from Vederiana, [a hamlet in Illyria,] men who at** home had to struggle incessantly against conditions of p**overty and all its attendant ills, in an effort to better their condition set out to join the army. [3] And they came to Byzantium, walking on foot and themselves carrying cloaks slung over their shoulders, and when they arrived they had in these cloaks nothing more than toasted bread which they had put in at home; and the Emperor enrolled them in the ranks of the soldiers and designated them for the Palace Guard. For they were all men of very fine figure.”

“[17] Such an Emperor had the Romans in Justinus. And he had a wife named Lupicina who, as being a slave and a barbarian, had been concubine of the man who had previously bought her. And she as well as Justinus attained the throne in the closing years of life.” Procopius, Secret History, 6.2-3, 17. Translated by Dewing, Loeb Classics, 1935. http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Procopius/Anecdota/6*.html

Procopius’ image of Justin is pretty clear: to improve his life, he joined the army and luckily made palace guard. The passages about them walking on foot, carrying cloaks, etc is to hammer home just how few means these men had. It may also be to underline Procopius' own distaste for him having come from such a background. When he finally gained the throne (I’ll talk more about this in question 3), even his wife was a barbarian-slave. There is really nothing glamorous about Justin in Procopius’ view. Unfortunately, there are no other accounts of his upbringing to compare against. Still, scholarship tends to think that Justin’s rusticity here as been embellished particularly due to Procopius’ criticisms of his nephew, Justinian (Cameron 2001, 63, n.1).

In sum, it is probably the case that Justin came from a more rustic background, but we should not overstate it (cf. Diocletian who also came from the lower-echelons of society).

2) Was he Illiterate? In short, no. Still, let’s take a peak at what Procopius says:

“For the Emperor Anastasius appointed [Justin] Commander of the Palace Guards. And when the Emperor departed this life, he himself, because of the power of his office, succeeded to the throne, being already an old man tottering to his grave, who had never learned to tell one letter from another, and was, as the familiar phrase has it, "without the alphabet," a thing which had never happened before among the Romans. And while it was customary for the Emperor to affix letters in his own hand to all documents containing the orders that issued from him, he was unable either to issue orders himself or intelligently to share in the knowledge of what was being done. But the man who drew the lot to sit as his Counsellor, Proclus by name, who held the office of Quaestor, as it is called, himself used to attend to all matters with independent judgment. But in order that they might have evidence of the Emperor's hand, those who had this matter in charge devised the following plan. Taking a small strip of prepared wood, they cut into it a sort of pattern of the four letters which mean in the Latin tongue "I have read," and dipping the pen into ink of the colour which Emperors are wont to use in writing, they would put it into the hand of this Emperor. And placing on the document the strip of wood which I have mentioned and grasping the Emperor's hand, they moved it and the pen along the pattern of the four letters, causing it to follow all the winding lines cut in the wood, and then went their way, carrying that kind of writing of the Emperor,” Trans. by Dewing, Proc., SH, 6.11-16; cf. Pseudo Zacharius viii.a [60-61] in Greatrex 2011, 279, and John Malalas, 17.1).

The account shown here is pretty vivid. Bold lines are essential information. The image we get here is Justin was essentially senile and hopelessly illiterate. He was so old, Procopius says, that his aids drew lots to see who would essentially be his hand maid, pun intended, to literally guide his hand for signatures! The episode is way too negative to be true. One must also ask, how on earth would one as hopelessly illiterate ever become commander of the palace guard?

Therefore, the assertion that Justin was illiterate is just false (Vasiliev 1950, 82-5; Baldwin 1989; Schlange-Schöningen 1995, 32–3; Rosen 1999, 777; Croke 2007, 18; Greatrex 2011, 280, n. 3). Theodoric the great is also called illiterate; thus, scholars tend to think that ‘illiterate’ probably just meant they lacked culture. Closer to home for me, Jordanes, the author of Getica and Romana, also calls himself illiterate, yet much of scholarship (though not all) tends to think he is just being humble (Getica, 265 ‘agrammatus’). All these individuals were contemporaries, an interesting detail. Finally, Brian Croke, 2007, really stresses Justin’s competence during his reign.

Thus, we can comfortably conclude that Justin being illiterate has most certainly been embellished and is not likely true at all.

edits: formatting & spelling (some serious formatting issue i can't fix. sorry.