Achilles is the focal point of Homer's Iliad, written in Greek in the latter half of the eighth century B.C.E. Seven centuries later, Virgil, writing in Latin, composed the Aeneid, whose composition ended in 19 B.C.E. when the poet died. Homer heavily influenced Virgil and Virgil's main character, Aeneas, has much in common with Achilles.
Both heroes have a mortal father (Peleus for Achilles; Anchises for Aeneas) and an immortal mother (Thetis for Achilles; Venus for Aeneas). We should note that Venus was a very important divinity for the city of Rome, of which Aeneas would become one of the founders.
Both heroes also experience the death of an ally at the hands of a key enemy. In Iliad 16, the Trojan Hector kills Patroclus, who is Achilles' best friend. In Aeneid 10, Turnus kills Pallas, who is an ally of Aeneas.
Both Achilles and Aeneas also avenge the deaths of this key ally. In Iliad 22, Achilles kills Hector; in Aeneid 12, in the closing lines of the epic, Aeneas kills Turnus.
We should also note that both Hector and Turnus were wearing military garb acquired from the person that they had killed. This military garb also affected the hero's mental state. Hector had taken Achilles' armor from Patroclus, who had borrowed the armor in an effort to trick the Trojans into thinking that Achilles had returned to the fighting. Turnus was wearing a belt that he had stripped from the body of Pallas. Aeneas saw this belt after he had Turnus down on the ground and Turnus was pleading for his life. The sight of this belt seems to have convinced Aeneas not to spare Turnus:
As soon as his eyes took in the trophy, a memory of cruel grief,
Aeneas, blazing with fury, and terrible in his anger, cried:
‘Shall you be snatched from my grasp, wearing the spoils
of one who was my own? Pallas it is, Pallas, who sacrifices you
with this stroke, and exacts retribution from your guilty blood.’ (A.S. Kline translation)
Finally, we should note that Aeneas' motivation in the epic is the establishment of a city that will eventually become Rome, the heart of the Roman empire. Achilles, in contrast, is motivated by personal glory. Aeneas has more nationalistic motivations.