Duels have a huge and long tradition. A while ago I wrote about the most unusual duel in history, which occurred in Paris in 1808.
Classical Greeks didn't have duels. Or if they did, it didn't make the histories in any significant way. I can't think of any, off-hand. Classical Greeks were much more into plotting and backstabbing.
The Iliad on the other hand is chockablock full of duels. After the fall of Minoan civilization the whole region went into a Dark Age. (Not the Dark Age we know, but an earlier one.) Duels were all the rage in that period, which also happens to be when Homer's stories come from.
The typical arrangement was that armies would line up, and then various champions would take on each other in individual combat before the general slaughter began. There's every reason to believe this was what happened in real life, but the majority of duels we know about occurred before the walls of Troy. The most famous is when Achilles slew Hector.
The most interesting I think occurred when Penthesilia fought Achilles.
Penthesilia was the daughter of Ares the God of War and Otrere the Queen of the Amazons. With that genetic heritage, a wise person would avoid annoying her.
Penthesilia accidentally killed her own sister in a hunting accident. In a fit of remorse, probably seeking honorable death, she presented herself to King Priam of Troy, who at that moment was sorely troubled. His son Hector had just died.
Penthesilia took the field, representing Troy. She slaughtered a whole pile of Greeks before coming up against Ajax. The fight against Ajax ended in a draw. Ajax went back to camp and told Achilles about the woman who was mowing down Greeks.
Achilles entered the fray and, inevitably, there was a duel.
This didn't end so well for Penthesilia. Achilles struck her in the chest and she fell.
A later writer named Propertius adds that after he killed her, Achilles raised Penthesilia's helmet to look upon her face, and instantly fell in love with her. Which was a trifle awkward since she was dead.
Though they didn't duel themselves, the classical Greeks were very keen on the Homeric combats, and interestingly, there are a lot of vase paintings showing Penthesilia vs Achilles. For some reason she doesn't seem to get the same airplay in modern retellings.