How did a Roman emperor become Christian and why did it cause the downfall of the Roman Empire?

It's easier to answer the second part of this question first: It didn't. But the spread of Christianity throughout the Roman Empire was closely influenced by the changing conditions, and influenced them in turn.

It's useful to break Roman Imperial history into a few time periods. The "Classical" Empire can roughly be considered to run roughly from the time of Augustus through the time of Hadrian – from 27 BCE to 138 CE. During this time, the Empire was rapidly expanding, funding each successive conquest from the spoils of previous ones. This was also the period of the earliest emergence of Christianity: Paul, for example, was traveling across Europe in the 40's and 50's CE, and died ca. 67 CE, right around the time that the first semi-standard Gospels began to circulate. Churches in this period were growing from small groups of believers with common interests to groups large enough to have schisms and arguments (as illustrated by Paul's letters, inter alia), but were not politically particularly significant.

The period between Hadrian and Diocletian (who took power in 284 CE) is sometimes referred to as the "Time of Troubles." Very broadly, Roman power had reached the extreme of its logistical capabilities, and there were no feasible conquests available to bring in additional wealth; and the broader Empire, which had been built up primarily to support this, didn't have any deep financial or transport logistics to deal with problems like climate shifts, crop failures, plagues, and (especially after 235) civil wars. This is an incredibly lengthy subject in its own right; my favorite place to start, if you want to learn more, would be Peter Brown's The World of Late Antiquity.

What's most significant is that the emperors of the post-284 world – referred to as Late Antiquity – faced a fundamentally different world than the ones of the Classical world. In particular, the Classical world had a concept of "Romanitas," of the different levels of being part of the Roman world, from foreigner to full citizen, and an entire social and religious universe predicated on the idea that Romans' relationship to the rest of the world had a lot to do with conquering it. (For example, there was the religious idea of the "Pax Deorum," the "Peace of the Gods;" prior to invading an area, it was custom to make sacrifices and offer a deal to the local gods, that if those gods would turn this place over to the Romans, the Romans would then build a temple to them in Rome. And build they did! This had important social effects: it signaled to people in Rome that this new area was now part of the Roman world, and could be enveloped within their systems, while signaling to the newly-conquered "clients" part of the ways in which they could be integrated into the Roman system in turn. If you're interested in the relationship of conquered people to the Romans, I highly recommend Ken Harl's lecture series for the Great Courses titled "Rome and the Barbarians." I actually recommend pretty much all his lectures because he's an amazing lecturer.)

In Late Antiquity, this old model of being a Roman no longer worked, and there was a palpable need for something to replace it.

Coincidentally, during the period of the Troubles, a lot of classical religious observance had waned. This wasn't a coincidence; classical Rome had little analogue to the modern idea of "faith" as a personal experience. Instead, religio – formal religious practice – was a social act, part of the formal rituals which bound society, and any type of personal belief was considered "superstitio" and somewhat suspect. (cf Beard et al's "Religions of Rome" v. 1 if you want a lot more on this) The decline of social structures quite naturally led to the decline of those religious practices. But in parallel, ideas of personal religiosity rose up, and there were three sets of religious ideas making major inroads: the worship of Mithras (a warrior god, very popular among the Army; we know relatively little of the details today, because a key part of the religion was its secrecy, open only to its converts), the worship of Magna Mater (basically a spread of the Egyptian idea of Isis into the wider Roman world; Egypt was the "mysterious foreign land of magic" to Romans, bolstered by everything from its magical-looking hieroglyphs to its long-standing industry of religion as a major industry), and Christianity. These three ideas were acquiring increasing political power as older sources of power collapsed, and starting to squabble amongst themselves rather intently. (cf Apuleius "The Golden Ass," arguably the first novel ever written, for an example; Apuleius was a partisan of Magna Mater and his characterizations of members of other religions is, shall we say, not subtle.)

Diocletian attempted some extremely profound reforms, such as partitioning the Empire into two more administerable halves (successful: the West collapsed completely a few hundred years later, but the East evolved into Byzantium), appointing an Emperor (the "Augustus") and a sub-Emperor (the "Caesar") for each, and setting a 20-year term limit (!) for Emperors, after which the sub-Emperor would ascend to the Purple. This happened exactly once, when Diocletian retired and forced his co-Emperor Maximian to do the same. Almost immediately, civil war broke out – the "Wars of the Tetrarchy." (The Wikipedia page is as good a place as any to start if you want to learn the bloody details) The key point in this war was the Battle of the Milvian Bridge outside Milan in 312, when Constantine (son of the Augustus of the West who succeeded Diocletian) defeated his chief rival Maxentius.

Historical accounts of exactly how this tied to religion, shall we say, differ a lot. The "official" account comes from Eusebius' "Life of Constantine," a few hundred pages of somewhat tedious hagiography, in which the story is that Constantine had a dream prior to the battle in which he saw a chi-rho (the main symbol of Christianity at the time) and was told "In this sign will you conquer!" ("In hoc signo vinces"). He thus painted this on his shields, converted to Christianity, got blessed by some handy bishops, immediately won, and converted the Empire.

More sober estimates concur as to a few points: (1) Constantine's mother appears to have been Christian. (Not uncommon; the Christians at the time focused on converting women, who would in turn convert their sons quietly, rather than men) (2) Constantine painted some symbol on various shields in the run-up to this battle, which was later interpreted (with his consent) as a Christian one. (3) There were definitely bishops traveling with him. (4) In 313, Constantine issued the Edict of Milan, which legalized Christianity. (5) Constantine was friendly to Christianity during his life, and actively interested in it; in 325 he convened the Council of Nicaea, to formally determine what Christianity "was." (6) For this reason, it was politically useful to become Christian during Constantine's reign, and so a lot of powerful people converted.

NB, 313 legalized, not officialized, Christianity: prior to this, Christianity had been illegal in Rome, for the nominal reason of refusal to sacrifice to the Emperor. (However, the accounts of widespread persecutions of Christians in the pre-Constantine period appear to be greatly exaggerated by later writers for political effect.) Constantine maintained a policy of ambiguity about his own religious leanings throughout his life, never openly declaring himself to be a Christian. (Something Eusebius tries to work around by claiming a deathbed conversion; nobody else attests this)

So this takes us to Christianity being legal and politically useful, and to a general public urge for a new Roman identity. Over the next century, Emperors were almost all explicitly Christian. (With the notable exception of Julian, a tremendously interesting figure!) On the political side, official support for Christianity steadily strengthened, including not just funding and quasi-official status, but turning a blind eye to things like bands of "destroyer monks" (another fascinating subject in its own right, and the seed of a thousand-year conflict between rural and urban religious power centers in Europe) ravaging the countryside, killing and destroying pagan artifacts, and ultimately laws against non-Christian practice. (To learn more about this, I highly recommend MacMullen's "Christianizing the Roman Empire, AD 100-400.")

(Continued...)