This is a slightly complicated one, and it unfortunately does not have a simple answer. The earliest attested indication of the Romans viewing themselves as the descendants of Aeneas (a Trojan prince) is in the histories by Quintus Fabius Pictor. He was familiar with the legend and it was popular by then, as his history starts with Aeneas arriving in Italy. He was writing circa 200 BCE. From here, it gets 'washier'. It is known that he utilized the writings of a Greek historian named Diocles. Diocles apparently wrote a history of Rome in the 4th or 3rd century BCE but it is lost, and we do not know all of what was taken from his history and what was added by Quintus or other later historians.

At the very least, we have a tradition of Romans viewing themselves as Trojan descendants by the 2nd century BCE if not earlier. Granted, this connection was not fully developed or extensively explored until Virgil wrote the Aeneid between 15 and 30 BCE.

EDIT: Grammar and spelling