Procopius is a good source, but he is just one man and he is not representative of contemporary attitudes. For one thing, we have no idea what kind of a man he was and what his intentions were. Each of his books were written very differently and with a different audience in mind, so how can we be sure that his account is accurate? Procopius' description of the plague for instance is very similar to Thucydides' account of the plague affecting Athens, so to what extent was Procopius writing an accurate eyewitness account of the events? He may very well have shaped his account to conform to the classical style in order to appeal to his audience. I would also disagree with /u/ApuleiusBooks' view that he did not attribute it to the supernatural or the evil of mankind, just take a look at this:

During these times there was a pestilence, by which the whole human race came near to being annihilated. Now in the case of all other scourges sent from Heaven some explanation of a cause might be given by daring men, such as the many theories propounded by those who are clever in these matters; for they love to conjure up causes which are absolutely incomprehensible to man, and to fabricate outlandish theories of natural philosophy, knowing well that they are saying nothing sound, but considering it sufficient for them, if they completely deceive by their argument some of those whom they meet and persuade them to their view. But for this calamity it is quite impossible either to express in words or to conceive in thought any explanation, except indeed to refer it to God.

I think it would be quite reasonable to say that Procopius blamed Justinian for the calamities of his reign. Procopius was many things, but he was not cold and rational (nor would we expect him to be), especially when he was free to let loose his rage in his Secret History:

All these cities were overthrown at that time by an earthquake, during which nearly all their inhabitants perished. Afterwards the plague (which I have spoken of before) began to rage, and swept away nearly half the survivors. Such were the disasters that afflicted mankind, from the day when Justinian first commenced to manage the affairs of the kingdom to the time, and after he had ascended the Imperial throne.

and

As for the plague, of which I have made mention in the former books of my history, although it ravaged the whole earth, yet as many men escaped it as perished by it, some of them never taking the contagion, and others recovering from it. But no human being in all the Roman Empire could escape from this man, for he was like some second plague sent down from heaven to prey upon the whole human race, which left no man untouched.

Lastly, Procopius was not the only witness, as we also have the accounts of John of Ephesus (a bishop) and Evagrius (a church bureaucrat). The former clearly attributed the plague to God and blamed the sins of mankind for causing it, calling it 'God's just sentence', whilst the latter wrote sadly of the many deaths in his family and how the plague struck at random, the voice of a man who was almost used to the grim cycles of death affecting him and his city (the plague came back at regular intervals):

And so at the outset of this great misfortune I was a¡ected by what are called buboes while I was still attending the elementary teacher, but in then various subsequent visitations of these great misfortunes I lost many of my o¡spring and my wife and other relatives, and numerous servants and estate dwellers, as if the indictional cycles divided out the misfortunes for me. Thus as I write this, while in the 58th year of my life, not more than two years previously while for the fourth time now the misfortune struck Antioch, when the fourth cycle from its outset had elapsed, I lost a daughter and the son she had produced, quite apart from the earlier losses.

Notably, all three accounts were written many years after the plague, so we don't have their immediate reactions to the events they lived through, but given what we have, I think it is quite easy to see that they did not understand the plague and attributed the deaths to divine justice, which isn't unexpected.

John of Ephesus did talk about his journeys from Egypt to Constantinople as the plague was spreading, so it's probably the best source to answer your question, though this passage still has its problems, as John wanted to emphasise the sufferings of the world to get people to live godly lives again:

At the same time that in the region of the capital these things were as yet known (only) by rumour, since they were still remote, and also before the plague (reached) Palestine, we were there. (Then) when it was at its peak we went from Palestine to Mesopotamia and then came back again when the chastisement reached there also, as well as (going) to other regions- Cilicia, Mysia, Syria, Iconium, Bithynia, Asia, Galatia and Cappadocia, through which we travelled in terror (on our way) from Syria to the capital (during) the height of the plague. Day by day we too - like everybody - knocked at the gate of the tomb. If it was evening we thought that death would come upon us in the night, and again if morning had broken, our face was turned the whole day toward the tomb. In these countries we saw desolate and groaning villages and corpses spread out on the earth, with no one to take up (and bury) them. [Followed by a long list of terrible things he saw on the road to the capital]

God's providence informed (us) about it in such a way that (news) was sent to every place in advance, and then the scourge arrived there, coming to a city or a village and falling upon it as a reaper, eagerly and or three miles (from it). And until what has been ordered against (one city) had been accomplished, (the scourge) did not pass on to enter the next. In this way it laid hold on (cities and villages) moving slowly. This is what (also) happened to this city: the visitation came upon it after (the city) had been perceiving the of the visitation by hearsay from all over the place for one or two years; (only) then did it reach (the city). swiftly, as well as upon other (settlements) in its vicinity, up to one, two years; (only) then did it reach (the city).

Procopius of Caesaria describes the Justinian plaque in clinical detail, noting in his "History of the Wars" how it spread from port to port. It is so accurate a description that it has a minor place in diagnostic history. Unfortunately he did not recognize that it was spread by fleas on rats. That said, neither did he particularly attribute it to the supernatural or the evil of mankind. I quote the bulk of it in my book "Antonina, A Byzantine Slut." Also, here is a link to an article about it (with quotes): http://hefenfelth.wordpress.com/2011/07/31/procopius-account-of-the-plague-in-constantinople-542/