Ranking the heroes on this list is hard. Achilles was the greatest hero to the Greeks in the Trojan War, but Hercules overcame so many obstacles that it is hard to rank him second to any. Besides, in Iliad I Nestor says that all the heroes of the Trojan War are second rate compared with earlier ones like Theseus. This list is mine, however, and 21st century prejudices are in play, so Achilles supersedes Theseus. In this list I will show the major claim to fame of each of these heroes.
For fun, try my quiz: Which Greek Hero Are You?

Public Domain. Courtesy of WikipediaHercules was outwitted by his nemesis
Hera from before the time he was born. After his birth she tried to destroy him by putting snakes in his crib, but he was too powerful even then. He grew up to become the hero who beat the odds time and again, performed amazing, often helpful tasks, and yet was full of human foibles, even to the point of having to expiate the horrible crime of murdering his own children (under the guidance of Hera, of course).

Clipart.com
Public Domain. Courtesy of Wikipedia.Theseus was the hero of the Athenians who saved them from having to send human sacrifice to the
minotaur. He volunteered to feed the half man-half bull beast in the labyrinth. With the help of the creature's half sister, Theseus was able to slay the monster and find his way out again.

Marie-Lan Nguyen/Wikimedia Commons.
Clipart.comPerseus was the son of Danae and Zeus. For a hero, having Zeus for a father was not unusual, but Zeus impregnated his mother as a shower of gold. After a bleak childhood, Perseus was helped by the gods to obtain the deadly head of snaky-tressed
Medusa. With the help of her head he could then turn anyone he wished to stone.

Public Domain. Courtesy of Marie-Lan Nguyen/Wikimedia Commons.Jason was another hero with an unfortunate childhood. To win back the throne he rightfully should have had, he was sent to capture the
golden fleece. He set out with a band of the greatest heroes the world has ever known and had a series of adventures worthy of epic. Eventually, with the help of the witch Medea, he took the fleece.
Jason's Angels

Creative Commons - Marsyas at Wikipedia.Bellerophon is a little short of status among the heroes. He rode on the winged horse Pegasus; killed the Chimera monster, and tried to fly Pegasus to Olympus.

Public Domain. Courtesy of Wikipedia.Orpheus is a hero for two reasons: he was counted among them in the Golden Fleece quest of the Argonauts and he survived a quest that even the more obviously heroic Theseus failed. Orpheus went to the Underworld to retrieve his wife, Eurydice, who had just died of snakebite. He made his way to the Underworld's royal couple, charming all in his passage with his extraordinary musical skill. Then Orpheus persuaded Hades to give him a chance to bring his wife back to life. He got permission -- on condition that he not look at Eurydice until they reached the light of day. While Orpheus failed to accomplish his mission, he returned to life without need of human intervention.

PD Courtesy of Bibi Saint-Pol at Wikipedia.Cadmus was the Phoenician founder of Thebes. He had set off on his adventures to find his sister Europa, but failed. When he settled down in Boeotia, in fulfillment of an oracle, he lost his men to a dragon (serpent) of Ares. Cadmus killed the dragon, planted its teeth and watched as armed men (the Spartoi) emerged from the ground. They fought each other down to a final 5 who helped Cadmus found Thebes. Cadmus married Harmonia, daughter of Ares, but Ares wasn't finished punishing the family of Cadmus for killing his dragon.

PD Courtesy of Bibi Saint-Pol at Wikipedia.Greek heroes were overwhelmingly men. In this top 10 list, the one noteworthy female athlete deserves a place. She is Atalanta, the sole women who sailed with Jason on his quest for the golden fleece, according to some. This paved the way for the return trip with the witch Medea of Colchis on board. Atalanta is also the hunter who first pierced the Calydonian Boar. Atalanta is best known, however, for losing a race because she kept stopping to pick up golden apples. As a result of losing the race, she was obligated to marry Melanion (or Hippomenes).