George Orwell
In this first person memoir, Orwell holds the central role. He explicitly defines himself in the opening of the piece as being a young police officer who despises the British imperial project in Burma, sides with the Burmese, and yet still feels that he has to prove his authority to the Burmese. As he explains this, he also states that at the time, he was too young to understand this about himself. We thus seem him play out exactly this inner conflict over the course of the story.
The sub-inspector
The sub-inspector is a Burmese policeman who answers to Orwell (and to the British). He's waiting for Orwell at the bazaar where the elephant has been rampaging.
The dead man
The dead man isn't so much a character as a figure—a body trampled into the mud by the rampaging elephant. He nonetheless plays in to the story as Orwell considers his own potential fate if he fails to properly shoot the elephant and merely makes it mad.
