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.
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")
During the Neutral Zone Incursion of 2266, Captain James T. Kirk identified the Achilles heel of a Romulan Bird-of-Prey that had been attacking the USS Enterprise as their need to disengage their cloaking device in order to launch their weapon, as it consumed all their energy. He planned to fire phasers at them when they dropped it. (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")
In 2367, Beverly Crusher described the Borg's interdependence as their Achilles' heel. She suggested that removing Jean-Luc Picard from their collective consciousness was akin to asking someone to disconnect one of their own limbs. (TNG: "The Best of Both Worlds, Part II")
Later that year, Q was delighted to discover Picard's Achilles heel; the Human emotion of love. He told Picard that if he'd known about this vulnerability which Vash had managed to unearth, he'd have appeared to the captain as a female instead. (TNG: "Qpid")