Since the original run, Shaffer has extensively revised his play, including changes to plot details; the following is common to all revisions.
At the opening of the tale, Salieri is an old man, having long outlived his fame. Speaking directly to the audience, he claims to have used poison to assassinate Mozart, and promises to explain himself. The action then flashes back to the eighteenth century, at a time when Salieri has not met Mozart in person, but has heard of him and his music. He adores Mozart's compositions, and is thrilled at the chance to meet Mozart in person, during a salon at which some of Mozart's compositions will be played. When he finally does catch sight of Mozart, however, he is deeply disappointed to find that Mozart himself lacks the grace and charm of his compositions: When Salieri first meets him, Mozart is crawling around on his hands and knees, engaging in profane talk with his future bride Constanze Weber.
Salieri cannot reconcile Mozart's boorish behaviour with the genius that God has inexplicably bestowed upon him. Indeed, Salieri, who has been a devout Catholic all his life, cannot believe that God would choose Mozart over him for such a gift. Salieri renounces God and vows to do everything in his power to destroy Mozart as a way of getting back at his Creator.
Throughout much of the rest of the play, Salieri masquerades as Mozart's ally to his face while doing his utmost to destroy his reputation and any success his compositions may have. On more than one occasion it is only the direct intervention of the Emperor himself that allows Mozart to continue (interventions which Salieri opposes, and then is all too happy to take credit for when Mozart assumes it was he who intervened). Salieri also humiliates Mozart's wife when she comes to Salieri for aid, and smears Mozart's character with the Emperor and the court. A major theme in Amadeus is Mozart's repeated attempts to win over the aristocratic "public" with increasingly brilliant compositions, which are always frustrated either by Salieri or by the aristocracy's own inability to appreciate Mozart's genius.
The play ends with Salieri attempting suicide in a last attempt to be remembered, leaving a confession of having murdered Mozart with arsenic. He survives, however, and his confession is met with disbelief, leaving him to wallow once again in mediocrity.

Question: Why does Salieri claim to have poisoned Mozart?
Answer: To earn fame.
Alternative answer: He wanted this to be something people would remember him by.