The Mahabharata gives Duryodhana arguments, and lets him make them at length. His father was the elder brother and was passed over for blindness, so on a strict reading of succession the throne came to the younger line by an accident of eyesight rather than by right. The Pandavas were fathered by gods rather than by Pandu, which he says out loud when it suits him. And they were presumed dead for years, during which he was the heir in fact.
His grievance is also personal and the epic makes that plain. Invited to Indraprastha for Yudhishthira's royal consecration, he walks into a pool of water he takes for a polished floor, falls in, and is laughed at by Draupadi and the servants. The dice game is proposed shortly afterwards. The epic does not offer this as a justification, but it does offer it as a cause, and it declines to make him hate without a reason.
At the end he refuses the peace terms that would have given the Pandavas five villages, saying he will not yield as much land as fits on the point of a needle. Dying of the broken thigh he claims the better position: he has lived as a king, given generously, kept his friends, and dies on the field, while his enemies have won by a series of tricks. The poem then does something startling. When Yudhishthira reaches heaven at the very end, Duryodhana is there, seated in honour, and the sight of him is one of the last tests the eldest Pandava is put through.