Yes, "Achilles" can have an apostrophe when indicating possession, as in "Achilles' heel." This usage signifies something belonging to Achilles. However, when referring to the character or name alone, no apostrophe is needed.
If you are indicating possession (Achilles' heal) use an apostrophe at the end of the word. If you are simply stating his name, there is no apostrophe.
Achilles
No. A spider is a spider and an apostrophe is an apostrophe.
Odysseus did not kill Achilles. Paris killed Achilles.
The Shield of Achilles
you've is the apostrophe of you have
The apostrophe for "they had" is "they'd".
The goddess of Achilles is actually the "Mother" of Achilles and the answer is Thetis (The Mother of Achilles)
Achilles often means the Achilles tendon, at the back of the heel.
No, your doesn't have an apostrophe. You're, however, does have an apostrophe because it's a contraction for you and are.
Patroclus used Achilles armor to fight Hector when Achilles refused to continue fighting for Agamemnon. Patroclus was Achilles close friend and lover in the Iliad.
Calcaneal (Achilles) Tendon