In fleshing out the early history of the Homeric heroes Achilles and Patroclus, Madeline Miller is following a tradition almost as old as the Iliad itself. Classical literature is full of the what-ifs and what-nexts of the Homeric stories. Aeschylus's play Agamemnon, for example, is an expansion of the brief account in the Odyssey of the king's murderous homecoming after the war. Euripides's Troiades is a brutal sequel to the stories of the women Helen, Cassandra, Andromache and Hecuba, the four who receive Hector's slaughtered body at the end of the Iliad. The poems have peculiar qualities that invite such expansions. The in-the-moment brightness of the text, the direct swiftness of the narrative, the open-endedness, the spareness: there is space for the imaginative reader to fill with backstory and sequel.

The past year has seen an outpouring of such Homeric reimaginings and fillings-out, such as David Malouf's novel The Ransom, based on book 24 of the Iliad; and Zachary Mason's Calvino-esque sequence of riffs on the Odyssey, imagining dozens of counter-fates for its central character. Now Miller, in her page-turning debut novel The Song of Achilles, brings us the boyhoods of Patroclus and Achilles. She is a respectful and clearly loving reader of Homer: nothing strikes a false note in her intricately created world at the court of Achilles' father Peleus, where ordinariness and wonder (centaurs, goddesses) are woven together without jerkiness. She nails her colours to the mast, too: Miller has her Achilles and Patroclus inseparably, gloriously and physically devoted, which certainly makes sense to this reader.

Where I lose her is when, instead of sticking to prequel, she forges on through the parts of the Trojan war described in the Iliad. Alas, the best of writers will stumble in comparison to the master. The scene in which Priam and Achilles meet in book 24 of the poem, the Trojan king stealing silently through the Greek camp to beg the hero for the corpse of his son, is one of the most moving, and dreadfully balanced, passages in all literature. There is no redemption here, nor forgiveness; anger pulsates beneath a surface gentleness. There is something else, too: an acknowledgment of shared suffering and shared humanity. But Miller, having started so well, has Priam say to Achilles, "Thank you for your hospitality", and "I am sorry for your loss". This is the bathetic stuff of the suburban dinner party.

With Chapman and Pope, and in the modern age Robert Fitzgerald and Robert Fagles, there is no shortage of English translations of the Iliad. Stephen Mitchell  who has also translated Gilgamesh, the Bhagavad Gita, the Tao Te Ching and the Book of Job  provides the latest (417pp, Weidenfeld & Nicolson, £25). He cites Matthew Arnold's advice to Homeric translators, to be "eminently rapid" and "eminently plain". "My intention," Mitchell writes, "has been to recreate the ancient epic as a contemporary poem."

To make familiar, or to make foreign? That is one of the many dilemmas of the Homeric translator. For my taste, Mitchell's version  while pacy and direct  is overfamiliar. The poems are grand; to use Arnold's description, they have nobility. I dislike Mitchell's technique of sometimes removing the poem's "Homeric epithets". (I mean the stock of often-repeated descriptive phrases, such as the "wine-dark" sea and Achilles "of the swift feet", which were probably used as metrically prefabricated units by the early oral bards who improvised these stories.) These phrases often simply fill out the metre and are irrelevant to the context, he argues. I think they do more than he gives them credit for. And in any event, I missed their delicious archaic tang. To me, they are one of the great pleasures of the poem.

Alice Oswald has stripped down the Iliad more radically, and more successfully, in her 80-page poem Memorial (96pp, Faber, £12.99). It is subtitled: "An Excavation of the Iliad", and described in her author's note as a "reckless dismissal of seven-eighths of the poem". Except it does not feel reckless at all, but precise and scalpel-sharp. The poem begins with a list of those characters killed in the course of the Iliad. This goes on for nearly eight pages. It is a catalogue of death, with the inscribed starkness of a war memorial. Some are just names; some, like Hector, the last name and the endpoint of Achilles' death-rampage, tug the heart.

The poem continues with Oswald wrapping words around each of these deaths; the words are Homer's, but refracted through her own lucent poetic imagination. She stitches into this unadorned fabric some of the glorious similes of Homer: those that imagine a peaceful, pastoral world away from the deathly field of war; or sometimes those that summon up something more dangerous. "Like fire with its loose hair flying rushes through a city / The look of unmasked light shocks everything to rubble / And flames howl through the gaps." As a reading of Homer, its ferocity of intent reminded me of Simone Weil's essay "The Iliad, or the Poem of Force". But it is also an exquisite and brutal thing taken entirely on its own terms. It's a major achievement.

If you were looking to retell the story of one of the great lovers in Greek myth, you could choose Paris, whose love (or lust) for Helen launched a thousand ships against his city of Troy, and wiped it from the map. Or Orpheus, whose devotion to Eurydice led him down into the depths of hell to try to reclaim her. Or even Alcestis, who loved her husband, Admetus, so much that she was willing to die in his stead.

The last mythical figure you might try to rework as a romantic hero would be Achilles, a one-man genocide whose defining characteristic was his unquenchable anger. Achilles withdrew from fighting at Troy because Agamemnon had slighted his honour, and watched his fellow Greeks being slaughtered by the Trojans; he only returned to battle when his friend Patroclus had been killed.

His revenge on Hector was merciless: not only did he kill the bulwark of Troy, he dishonoured the corpse, dragging Hector's body around the city three times. Homer sets all this out in the opening line of The Iliad: "Sing, muse, of the wrath of Achilles." Even after death, his ghost still thirsts for blood, and Polyxena, a Trojan princess, has to be sacrificed at his tomb before the Greeks can sail home from Troy.

But in her novel, The Song of Achilles, Madeline Miller has found the lover beneath the bloodshed and fury. The story is told from the perspective of Patroclus who, exiled by his father to live in the court of Peleus, soon falls in love with his host's son, the superhuman Achilles: from childhood, his demi-god status means he is swifter, more beautiful and more skilled than all his peers.

Astonishingly to Patroclus's eyes, Achilles returns his love, and the two boys grow into adulthood and a love affair. Achilles remains a godlike figure to Patroclus: "Then I turned to look at him. He was on his side, watching me. I had not heard him turn. I never hear him."

Miller's prose is more poetic than almost any translation of Homer. Iphigenia, sacrificed by her father Agamemnon to appease the goddess Artemis, is described as having "a tripping name, the sound of goat hooves on rock, quick, lively, lovely". And when Briseis, the concubine Agamemnon later steals from Achilles, gradually learns Greek from Patroclus, "her words were like new leather, still stiff and precise, not yet run together with use".

The sense of impending tragedy is never far away from these lovers. Achilles has long known that he must choose between a short, glorious life, and a long one lived in obscurity. Miller ramps up the dramatic irony inherent in their story. Both know Achilles will never return from Troy: he is fated to die there. But Patroclus is too obscure to figure in prophecies, so he dreads the horror of life after Achilles's death: "I rose and rubbed my limbs, slapped them awake, trying to ward off a rising hysteria. This is what it will be, every day, without him. I felt a wild-eyed tightness in my chest, like a scream. Every day, without him."

We know Patroclus must die before Achilles. And only once he is dead, does the truly terrifying aspect of Achilles's nature come to the fore. When he faces Hector, the latter asks that his body be returned to his family, when Achilles is done with him. "Achilles makes a sound like choking. 'There are no bargains between lions and men. I will kill you and eat you raw.'"

Miller spent 10 years writing this book, yet her smooth prose conceals the painstaking research she has clearly put into it. This is a deeply affecting version of the Achilles story: a fully three-dimensional man  a son, a father, husband and lover  now exists where a superhero previously stood and fought.