Alexander presumably had a number of helmets over the course of his life--certainly the images of Alexander presented by the coins of the Diadochi present him as wearing several different helmets, many with some kind of divine attribute. I must assume you're referring to the description of the helmet and armor Plutarch says he received when he visited Troy:

ὠσαμένων δὲ πολλῶν ἐπ᾽ αὐτὸν ἦν δὲ τῇ πέλτῃ καὶ τοῦ κράνους τῇ χαίτῃ διαπρεπής, ἧς ἑκατέρωθεν εἱστήκει πτερὸν λευκότητι καὶ μεγέθει θαυμαστόν

Then many rushed upon him, and indeed he was distinguishable by his shield and the crest of his helmet, on either side of which stood a wing, marvelous in size and whiteness.

A little later Plutarch tells us what happened to this helmet. In Alexander's heroic combat during the cavalry engagement at the Granicus the helmet was irreparably damaged:

καὶ τόν μὲν λόφον ἀπέρραξε μετὰ θατέρου πτεροῦ, τὸ δὲ κράνος πρὸς τὴν πληγὴν ἀκριβῶς καὶ μόλις ἀντέσχεν, ὥστε τῶν πρώτων ψαῦσαι τριχῶν τὴν πτέρυγα τῆς κοπίδος

and he struck off the crest at one of the wings, and the helmet barely and with difficulty held up to the blow, so that the edge of his sword touched the top of his hair

Spithridates, who delivered the blow, was struck down, says Plutarch, by Black Clitus, but the damage was done to Alexander's helmet. As for the rest of Alexander's armor, Arrian tells us what happened at Troy:

ἀνελθόντα δὲ ἐς Ἴλιον τῇ τε Ἀθηνᾷ θῦσαι τῇ Ἰλιάδι, καὶ τὴν πανοπλίαν τὴν αὑτοῦ ἀναθεῖναι ἐς τὸν νεών, καὶ καθελεῖν ἀντὶ ταύτης τῶν ἱερῶν τινα ὅπλων ἔτι ἐκ τοῦ Τρωικοῦ ἔργου σωζόμενα. καὶ ταῦτα λέγουσιν ὅτι οἱ ὑπασπισταὶ ἔφερον πρὸ αὐτοῦ ἐς τὰς μάχας.

[They say] that he went up to Ilium and sacrificed to Ilian Athena, and that he hung up his own equipment at the temple, and took down [votive offerings are said in Greek to be "hung up" in the temple] some of the holy armor preserved from the Trojan struggle. And they say that his shield-bearers [here οἱ ὑπασπισταὶ likely refers to his personal squires, not to the corps of Hypaspists] bore these in front of him in battle.

Peter Green conjectures, I think not unreasonably, that Alexander wore this equipment (from which he thinks Alexander's plumed helmet came) at the Granicus, but that it was so beat up there that in the future he just had it paraded in front of the army. In any case no text says anything about it being Achilles' armor (Achilles, in any case, being a fictional character means that no matter what he could not "really" have had his armor). I think you must be confusing the attestation in several texts that Alexander visited the supposed tomb of Achilles while at Troy--I don't know where this stealing business comes from, Alexander was notably respectful at Troy. Troy in Alexander's lifetime was sort of a little theme park, a site only sort of geographically related to the site of Troy VIIA (which in any case the Greeks didn't know about), and naturally it was the most notable location of "genuine" relics from the Trojan War, something many sanctuaries claimed to have.

There's not really much else to say. No portrait of Alexander preserves an image of this helmet, nor does any coin either from his lifetime or that of the Diadochi. This coin, minted first by Alexander and then under Philip Arrhidaeus, depicts Athena, not Alexander, with a helmet that seems to have a double plume, but I don't know of anyone who suggests that it's supposed to be a copy of Alexander's helmet. Indeed, Hellenistic portraits of Alexander either show him with overtly divine headgear (e.g. the Hercules lion skin or an elephant skin, probably associated with Dionysus), or have him bareheaded (e.g. the Alexander Mosaic). The Alexander Sarcophagus has him wearing a lion's scalp, and some modern illustrations combine this with Plutarch's description of the plumed helmet at the Granicus, but this is conjecture and not especially scholarly conjecture at that