Achilles was a character in Greek mythology. He participated in the Trojan war according to Homer. He was nearly invulnerable except for one small part of his body, which finally cost him his life. Such a minor vulnerability on an otherwise invincible enemy became known as an Achilles' heel.
During the Neutral Zone Incursion of 2266, Captain James T. Kirk referred to the term Achilles' heel upon explaining to Lieutenant Stiles that they had discovered the technical vulnerabilities of a specific Romulan Bird-of-Prey they were trying to defeat. (TOS: "Balance of Terror")
In 2365, Doctor Katherine Pulaski described Data as "sulking like Achilles in his tent," after the android developed a fixation with his own fallibility following a Strategema defeat. (TNG: "Peak Performance")
Leonardo da Vinci was contracted to paint a portrait of a local Cardinal's nephew under the conditions that the individual be depicted in the heroic mode of an Hercules or an Achilles. The end result was described as having made the "young fool of a nephew look far more heroic than nature ever intended." Da Vinci described the feat as "an act on my part far greater than anything accomplished by Hercules or Achilles!" (VOY: "Scorpion")
Henry Janeway once pointed out to his son, Jason Janeway, that none of the classic heroes had come from nuclear families – in fact, Hercules, Achilles and Odysseus were all products of single parent upbringings. (VOY: "11:59")