The Seven Deadly Sins as we know them were worked out by St. Gregory the Great. He was building on predecessors. Evagrios had a similar sort of list, as did St. John Cassian, but Evagrios's list isn't really the Seven Deadly Sins in the proper sense, first, because it is a list of eight; and second, because Evagrios is actually talking about kinds of temptation (distracting thoughts) rather than vices. Cassian could reasonably be considered the first person to speak of them (although he attributes them to Abbott Serapion) precisely in a form that makes them a list of vices, since he speaks of them as principal faults. And Conferences 5 makes an interesting, if extraordinarily complicated, set of arguments that will clearly inform the later tradition. But Cassian, like Evagrios, is concerned specifically with monastic life, and his list is not quite the same as the later list, although one can see that it's pretty close:
There are eight principal faults which attack mankind; viz., first gastrimargia, which means gluttony, secondly fornication, thirdly philargyria, i.e., avarice or the love of money, fourthly anger, fifthly dejection, sixthly acedia, i.e., listlessness or low spirits, seventhly cenodoxia, i.e., boasting or vain glory; and eighthly pride.
To get the actual traditional list of Seven Deadly Sins, we have to go to Gregory's modification of them in the Moralia in Job. Gregory generalizes Cassian's list, gives it a simpler and more straightforward organization, precisely characterizes why these are the principal faults, gives the order that will become standard, and does it all in an image so striking that it's not at all surprising that it grabbed everyone's imagination. You really need to read it for yourself (this is from Book XXXI). He's been talking about life as a heavenly soldier fighting for virtue, all very interesting, but then he hits Job 39:25, which talks about captains and a howling army, and he takes that and outdoes himself:
87. For the tempting vices, which fight against us in invisible contest in behalf of the pride which reigns over them, some of them go first, like captains, others follow, after the manner of an army. For all faults do not occupy the heart with equal access. But while the greater and the few surprise a neglected mind, the smaller and the numberless pour themselves upon it in a whole body. For when pride, the queen of sins, has fully possessed a conquered heart, she surrenders it immediately to seven principal sins, as if to some of her generals, to lay it waste. And an army in truth follows these generals, because, doubtless, there spring up from them importunate hosts of sins. Which we set forth the better, if we specially bring forward in enumeration, as we are able, the leaders themselves and their army. For pride is the root of all evil, of which it is said, as Scripture bears witness; Pride is the beginning of all sin. [Ecclus. 10, 1] But seven principal vices, as its first progeny, spring doubtless from this poisonous root, namely, vain glory, envy, anger, melancholy, avarice, gluttony, lust. For, because He grieved that we were held captive by these seven sins of pride, therefore our Redeemer came to the spiritual battle of our liberation, full of the spirit of sevenfold grace.
He goes on to note that each of these vices spawn many more vices (giving precise examples), that five are spiritual vices and two (gluttony and lust) are carnal vices, and gives a brief account of how each of these vices take root. In later lists, avarice, being a sort of borderline vice, sometimes gets treated as spiritual and sometimes as carnal. We do have here all seven of the Seven, and, what is more, we have them in the order that will become standard. Notice that there is a sense in which there are still actually eight principal vices: there's a super-vice, the queen of vices, which is pride, and pride is the source of the Seven, as her generals. This will be part of Aquinas's account, too, since Aquinas's account of the Seven is essentially Gregory's, but given a carefully laid-out philosophical backing. Aquinas's account will later become the major source on the subject. But in Aquinas's account the connection between pride and the Seven is not so clear as it is in Gregory, due to a different format, and is certainly not put in such a striking way. Because of this very late accounts of the Seven will sometimes conflate pride and vainglory; this simplifies things a bit, but also makes it unclear how the Seven are related to each other, whereas this is very clear in both Gregory and Aquinas. It also tends to trivialize pride, which as a vice was traditionally understood as something far and away more insidious, encompassing, and dangerous than vainglory (although vainglory, as being the worst and most dangerous vice springing from pride, has always had a special connection with its root). And it's important to recognize that each of the Seven, even lust, gluttony, and avarice, can be considered as an extension of pride, as a kind of pride in a broad sense. Once this sense is lost we begin to get a sort of an impoverishment of the way people understand the Seven. There are still some major and important contributions once this has begun to happen: Dante's Inferno and Purgatorio are based on them, and the Parson's Tale in Chaucer's Canterbury Tales (the 'tale' told as the pilgrims finally approach Canterbury and therefore need to get into the right frame of mind after their various vices have been put on display), is just a confession manual based on the Seven that is adapted into a sermon by added some funny touches as the pious but stern Parson speaks the truth but sometimes gets off track a little or unintentionally says something funny (my favorite point is in his discussion of pride, when he ends up ranting somewhat inordinately about the clothes kids wear which show off their backsides as if they were she-baboons, which is not quite what one would expect from a sermon on pride). But most of the contributions from this point end up being in striking images more than in substantive clarifications or improvements.