Who was the most successful empire: Athens, Persia or Macedonia? Why?

In asking this question, you first need to define by what metric you are defining successful. From that perspective, I can't really adjudicate between the three choices here, but I can give a basic outline differentiating the three empires in question in order to illustrate some of the contrasts between them.
Persia was the oldest of the three empires, and up to that point, the largest seen in world history. It was founded by Cyrus the Great, and at its peak would extend from Egypt and the Balkans in the west, all the way to India, controlling much of Central Asia. It was also the longest lasting of the three. Cyrus the Great reigned from 559 to 530 BCE, and the empire would continue until the conquests of Alexander the Great (r. 336–323 BCE). Politically, the Persian Empire has been much celebrated for its policies of toleration towards local religions and customs. It has also retained a lasting legacy, and its example would serve as an inspiration for later Empires in Persia and the Middle East.
The Athenian Empire grew in the shadow of Persian power. It actually originated as a defensive league against the threat of Persia, cobbled together after Xerxes' failed invasion of the Greek mainland (480–79 BCE). The Athenians later converted this defensive league into an empire, with Athens at the center. From here, the Athenians would use the empire to fuel the so-called Golden Age of Athens. However, they would ultimately be pulled into the Peloponnesian War against Sparta (431–404 BCE), which would result in the dissolution of the Athenian Empire. Of the three, it was the smallest by a wide margin, and relied heavily on Athenian naval power, being that it stretched across the coasts of Greece and Asia Minor, as well as across the islands of the Aegean Sea.
The last of the three is probably the trickiest, because so much of Macedon's history is tied up with the conquests of Alexander the Great. Ultimately, Macedonian power was largely built up by Alexander's father, Philip (r. 359–336 BCE), whose military innovations resulted in the conquest of the Greek city-states. After his father's assassination, Alexander would then use the foundation Philip had built to conquer the Persian Empire. His conquests would stretch all the way east into India. However, unlike Cyrus, he was unable to provide for a stable transition of power. After his death, his empire fragmented into the Successor States of the Hellenistic World.