There was nothing especially unusual in the Roman claim of Trojan descent. For one thing, ethnographies assigning Greek ancestry to various non-Greeks were a common feature of Greek literature already by the time large-scale contact with the Romans was a reality. For another, the story of Aeneas' flight to Italy was a Greek one, not a Roman one. It's possible that already in the late Dark Age the story of Aeneas' flight into Italy was known to Greek poets through the Epic Cycle, but certainly by the fifth century, long before the mainland Greeks knew much about the Romans at all (the first Greek mention of Rome was either Aristotle or Heraclides Ponticus writing about the sack of Rome in the fourth century) it was known. Dionysius of Halicarnassus mentions that Hellanicus of Lesbos (fifth century) said that Aeneas sailed west across the Hellespont into the Chalcidice, whence he (again, according to Dionysius on Hellanicus' authority) crossed the land of the Molossians with Odysseus (or after, the phrasing μετ᾽ Ὀδυσσέα is frustratingly ambiguous with the accusative) and came into Italy. There, says Dionysius' source, he founded the city of Rome after one of the Trojan women who had come with him, a certain Ῥώμη. The name means "power," and no doubt the Greek ethnographer's mind--always eager to find a folk etymology connecting back to Greek somehow--jumped at the correspondence between Greek and the Roman city. Aristotle, who knew of the existence of Rome but knew next to nothing about it, Dionysius says related a slightly different version. Instead of ascribing Roman origins to Aeneas, Aristotle claimed that some of the Achaeans were blown off course to Latium while returning from Troy. When their ships were burned by their Trojan captives, unwilling to be led into slavery, they were forced to stay in Latium. Thus Aristotle ascribed combined Greek and Trojan ancestry to the Romans. A little later, again on the authority of Dionysius, Callias wrote that Ῥώμη married Latinus, in an interesting inversion of the Roman tradition of Aeneas' marriage to Lavinia, and bore three sons: Romus, Romulus, and Telegonus. Callias, an early Hellenistic Sicilian, would certainly have had access to Italian traditions, and this version is obviously a synthesis of Odysseus' three children with Circe (whom Dionysius calls Romus, Anteius, and Ardeius, from Rome, Antium, and Ardea) and a native tradition of Trojan ancestry. These early Greek stories are a mess (even moreso than the Roman stories, which were not yet codified even under Augustus--Livy has a different version of the Aeneid) of native traditions mixed with Greek ethnographic attempts to tie the Romans to various Greek or Homeric groups. It's difficult to tell what originated where and when, but at the very least written versions of a story that Aeneas or some other Trojan founded the city exist in Greek literature by the fifth century. This is long before any Latin literature, but contacts between the Greek mainlanders and the Romans were still tenuous at best in this very early period. These stories did not come out of nowhere--the idea that the Romans adopted Trojan ancestry relatively late is not usually taken seriously. For one thing, archaeology seems to be generally against it. Scenes of Aeneas carrying Anchises are some of the most common images on Athenian vases in Etruria from the end of the sixth century. Moreover there's a very famous statuette, presumably votive in nature, from Veii around the end of the sixth century of Aeneas carrying Anchises. Stories of Trojan ancestry were also not limited to Rome. Likely Aeneas himself was transported to Latium either from Etruria or from Magna Graecia, but this certainly happened at a very early date from outside, and Aeneas was quickly assimilated with some kind of pater indiges figure, whose identity is hard to pin down--Aeneas had various titles throughout Italy related to this, including Jupiter Indigens, Pater Indiges, Deus Indiges, Aeneas Indiges, etc. Livy tells a version that Aeneas died at the river Numicus (mentioned in the Aeneid and in Ovid, wherein Aeneas is cleansed, deified, and renamed Indiges), a river of great ritual significance for the Latins (in Ovid Anna Perenna also dies in the same river). These stories originated very early and were closely integrated with the various origin stories floating around central Italy in the Roman Archaic Period--the Romans didn't adopt them all of a sudden, and they appear in Greek first