How did Athenian empire develop after its victory over Persia?

The victory over Persia in 480-479 was achieved by a southern Greek alliance - there was no Athenian empire.

Sparta proposed that Greeks located within the Persian empire, who had caused the problem in the first place, be repatriated back to mainland Greece. Athens proposed that an alliance of the cites sustain a fleet which could guarantee protection of the Greek cities within the Persian empire. This was implemented (the Spartan plan was adopted over 2,000 years later in 1923 CE when Greeks in Turkey were repatriated).

The cities contributed ships or money. Most took the easy option of money which allowed Athens to maintain its superior navy at little cost. It was rigorous in collecting the contributions, by force when necessary. While the league was not an Athenian empire, it has been called that in modern terms.

After further battles withe Persia in 466 and 450 BCE, a peace was agreed in the 449 BCE Peace of Callias. Athens continued to collect the tribute and used it to beautify the city (Parthenon etc) and pay for nearly half its citizens to on the public payroll.

This was a very cozy arrangement, however Athens went too far in getting involved in the Peloponnesian war, as a result of which it lost this comfortable empire, as well as wrecking the Greek world in the process.