The persian empire was eventually weakened by

King Xerxes tried to carry on his father Darius I's attempt to establish an ethnic frontier by subjugating mainland Greece, as the mainland Greek states were fomenting rebellion in the Greek states in Asia which were under Persian control. He lost, and Athens established an anti-Persian league which dominated the eastern Mediterranean. 

Although the internal fighting in the Greek world subsequently eroded the strength of the Greek states, and Persian gold was able to buy influence in them, the rise of Macedonia, and Philip II's and then Alexander's control of Greece tipped the balance of power against Persia. 

Alexander took the empire's Mediterranean coastline and so removed the Persian fleet, and as on land Persia's military forces were unable to match the power of Macedonian-Greek forces, he was able to defeat them in three major battles, and so take over the Persian Empire.