With the preface that I think it's silly upset about mythological accuracy in a TV show: since the Olympic gods neglected to photograph their human pets, we can't really know what Achilles looks like. Nor do the ancient epics give much in the way of physical descriptions. That said, literary sources do describe Achilles as having blonde or reddish-blonde hair.
From the Iliad, 1.197:
The goddess standing behind Peleus' son caught him by the fair hair
Statius in his Achilleid 1.158-162 likewise describe the hero as:
lo! he was come, made larger by much dust and sweat, and yet for all his weapons and hastened labours still pleasant to the sight; a radiant glow shimmers on his snow-white countenance, and his locks shine more comely than tawny gold.
Note that according to Hyginus in his Fabulae, Achilles was called Pyrhha when he was disguised as a woman, because the Greeks called redheads pyrrhos.
Hyginus added that, while Achilles was disguised, he was called Pyrrha (Red) because of his reddish yellow hair.
Brewer, Wilmon. Ovid's Metamorphoses in European Culture: Books VI-X. Cornhill Publishing Company, 1941.