Our sources for Achilles beyond Homer are not that great. A lot of what is familiar about Achilles in the modern mythos is from much later sources, like Statius (1st century CE, or c.800 years after the Homeric epics had been more or less composed). That means that most of his "biography" as found on the internet is a hodgepodge of sources ranging over a millennium of input. For example: his invulnerability and his weakness in the heel is not well-attested in Homer or contemporary sources.

Still, the vague outlines of his story come out. From birth he was prophesied (by Themis or Prometheus) to be greater than his father, or to overcome his father in some way. Zeus and Poseidon had been keen to get with Thetis, the sea nymph, but when they heard this prophesy concerning Thetis, they passed her off to a mortal (though Peleus was mighty among mortals).

He was educated by the famous centaur hero, Chiron. This centaur, in various versions of various myths, educated all the great heroes of Greek mythology (Theseus, Herakles, Jason, Meleager, Aeneas, Nestor, Diomedes, etc). He is also said (by Homer) to have grown up in the always-fun-to-pronounce Phthia along with his extra-special companion Patroklos.

There is a post-Homeric, but awesome, story in which Thetis tries to hide the young Achilles to protect him from being caught up in the Trojan War. He is said to live among the women on Skyros, dressed as a woman. In the story, Odysseus learns that the Greeks cannot win without Achilles, and he devises several tricks to make Achilles come out of hiding. While on Skyros, Achilles variously rapes a gal and/or fathers a son, Neoptolemos, also called "Pyrrhus" after his alias while on Skyros (Pyrrha, "red-haired girl").

Next is the convoluted and confused story of Achilles and Telephus. I won't get in to it here, but during a battle Achilles wounded Telephus, and the wound would not heal without the one who inflected the wound giving healing. So Telephus goes on an adventure to find Achilles and ask/trick for healing.

There is also a collection of story fragments pertaining to the Trojan War before the events of the Iliad begin. They all revolve around Achilles destroying nearby cities or attacking people/things. For instance, during this time he captures the place (I forget the name) where he acquires the girl Briseis as his war-prize, whom Agamemnon will later steal from him.

There are other stories, some more shaky than these as far as provenance.