The only source that mentions Nero's blaming of the Christians is Tacitus (Annals 15.44):

Therefore, to scotch the rumour, Nero substituted as culprits, and punished with the utmost refinements of cruelty, a class of men, loathed for their vices, whom the crowd styled Christians. Christus, the founder of the name, had undergone the death penalty in the reign of Tiberius, by sentence of the procurator Pontius Pilatus, and the pernicious superstition was checked for a moment, only to break out once more, not merely in Judaea, the home of the disease, but in the capital itself, where all things horrible or shameful in the world collect and find a vogue. First, then, the confessed members of the sect were arrested; next, on their disclosures, vast numbers were convicted, not so much on the count of arson as for hatred of the human race. And derision accompanied their end: they were covered with wild beasts' skins and torn to death by dogs; or they were fastened on crosses, and, when daylight failed were burned to serve as lamps by night. Nero had offered his Gardens for the spectacle, and gave an exhibition in his Circus, mixing with the crowd in the habit of a charioteer, or mounted on his car. Hence, in spite of a guilt which had earned the most exemplary punishment, there arose a sentiment of pity, due to the impression that they were being sacrificed not for the welfare of the state but to the ferocity of a single man.

The passage has caused a great deal of scholarly discussion over the years, including some doubts over its authenticity. However, the general consensus is that this is a genuine bit of Tacitus, and yes, that makes this pretty much the earliest mention of Christians as a distinct group by a non-Christian author. Suetonius and Pliny the Younger also mention the Christians, and both were writing in the early 2nd century, so the same time as Tacitus. In both of these writers, the Christians are also coming in for rough treatment by the Roman authorities - in Suetonius it's Nero again (and maybe Claudius a decade earlier), and in Pliny it's actually himself. He was the governor of the province of Bithynia, and oversaw the trials of some Christians. His letter describing his actions to the Emperor Trajan is a fascinating window into the interactions between the Roman state and the new religious group, which is less than a century old at this point, and probably only numbers a few thousand adherents across the whole empire. The crucial difference between the writers here is that Tacitus and Suetonius are describing events around 50 years before their time, whereas Pliny is talking about his own experience. How much does the contemporary situation colour the accounts of Tacitus and Suetonius, in terms of their awareness of Christianity as a distinct group? There's also genre and authorial intent to consider here, especially with Tacitus. It's always important to remember that throughout his Annals, Tacitus is intentionally presenting a gloomy, degenerate picture of Rome, fallen from its glorious past. His portrayal of Neronian Rome is a centrepiece to this, and we can't take the tiny passage about the Christians out of this context. Tacitus lumps Christianity in with the other horrible and shameful things that have collected in Rome under the emperors, but makes it pale in comparison with the cruel debauchery of Nero, who allegedly burned the Eternal City to satisfy his own desires and then tortured innocent people to cover his tracks.

As to how Nero became aware of Christianity, we can only speculate. Tacitus implies they were notorious in the city, and the evidence from the New Testament suggests a fairly strong community. Paul wrote his letter to the Christians in the city before he visited, and preached there for two years while awaiting trial. Perhaps someone in the imperial household (or even Nero himself) picked up gossip or rumour of these new beliefs, or maybe heard someone preaching on the streets. Pliny's letter suggests that perhaps the emperors might have also been getting news about the religion from other provinces, especially when Christians were caught up in (or the cause of) civil unrest.