It is always problematic when trying to adapt literary characters to the game, and with a mythological figure, it's especially so, since there is no single source to which one can appeal.
Let's lay out some basics, though:
The Greco-Roman sources for Achilles that survive include Homer (c. 700 BCE, reflecting an earlier oral tradition that preserves details from the Bronze Age) to tragedy (he's a character in Euripides' Iphigenia in Aulis, produced in 405 BCE) to Hellenistic works (including Apollonius' Argonautica, where he appears as a baby) to Roman works in Greek (Apollodorus' Library) and Latin (Ovid, Hyginus) that extend into the second century CE. That's 900+ years of history, plus ancient commentators on the lot, and not including lost works throughout that period but which are indirectly attested, that we refer to when we're talking about "the myth" or "the legend". Throughout that time, there is no obligation on any of these creators to adhere to what has gone before, and each will use the character in their own way for their own literary purposes. "The Myth" simply doesn't exist; there is only an ongoing accretion of detail some of which will agree with what's gone before, and some of which won't.
We can't know "the original story" (it predates literacy) and so if we want to speak at all meaningfully about this, we need to pick an author we like.
The OP mentions Homer and the
Iliad, but if you're going with that, you find yourself quickly stuck.
What better place to begin this series than the protagonist of the Illiad, slayer of Hector—that son of a nymph Achilles!
Homer doesn't know about the invulnerability. In fact, that detail isn't attested until after the death of Alexander the Great, four centuries later. Some scholars guess it's an earlier tale, but "the legend" and "the myth" simply doesn't support it.
Though not invulnerable, he is able to fight a river and then kill Hector (Iliad 20-22). For Homer, he demonstrates speed ("swift-footed Achilles"), high constitution and strength (he is the single best hand-to-hand fighter at Troy, on either side; so 20 Strength and Constitution?).
In the
Odyssey, he is the wisest person Odysseus encounters in the underworld, so we should give him at least a posthumous 18 wisdom.
Vase-painting (and lost tragedies about Palamedes) have him regularly playing tactical board games, which suggests Intelligence. Euripides in the fifth century plays up his Charisma. Etc.
In Homer, he uniquely uses his father's spear (which for some reason only he an use/lift). He doesn't use a bow or a sword (though there is a really good sword scene at the start of the movie Troy). [EDIT: He has a sword, which he almost uses against Agamemnon, in book 1.]
I understand the desire to stat up mythic figures, but (not to be too much of a pessimist) it's a mug's game, and you can justify almost any decision because the sources simply do not agree with one another.
The same will be true of other figures who are less well attested (Daedalus) or who developed over several hundred years (Lancelot, Robin Hood).
There is not, and cannot be, a right answer.