How did the Persian empire come about? What challenges did the rulers face and what institutions and policies did they devise to meet those challenges?

The Persian Empire occupied an area that is now mostly located in modern Iran. It originated in the late Bronze Age when the Median people migrated into an area of Northern Iran from Central Asia. They spoke an Indo-European language and took advantage of a temporary power vacuum caused by the decline of the  Middle Assyrian Empire.
From roughly the tenth through the seventh centuries, the Medes were ruled by the vast and powerful Neo-Assyrian Empire. In the late seventh century, civil wars and rebellions by subjects including the Medes, Babylonians, Chaldeans, Egyptians, Scythians, and Lydians led to the collapse of the Assyrian Empire. The Medes took advantage of this to conquer Nineveh and establish a Median kingdom or area of influence. Our information about this period, unfortunately, is limited. Our major source is the Greek historian Herodotus, who is not completely reliable.
In the seventh century, the Persians had settled in the southern part of the Iranian plateau. Also speaking an Indo-European language, they were nomadic peoples who had arrived in the tenth century. It was the unification of the Medes, Persian, and Parthians that led to the formation of the Persian Empire.
The Achaemenid Empire or First Persian Empire (c. 550–330 BC), also called the First Persian Empire, was founded in 550 BC by Cyrus the Great, inverting the power relationship in which previously the Medes had dominated the Persians. The major challenges he faced were administrative. He needed to transform a small, tightly-knit, ethnically uniform nomadic society into a multi-ethnic empire controlling a fixed territory. The Persians borrowed many administrative ideas from Mesopotamia.
One of the biggest problems in governing an ancient empire was communication from the center to the periphery. The Persian kings addressed this several ways. First, they built substantial road networks and a system of messengers. Cyrus was also responsible for supporting a comprehensive postal system. 
Next, local satraps were given the authority to govern specific regions of the Persian Empire and given considerable autonomy. This regional autonomy included religious and cultural freedom. 
The Persian kings sustained their rule economically with a taxation system that required each satrapy to raise a certain amount of taxes, which were used to support a professional army and maintain civil services. 
The Persian Empire had a uniform legal code and generally solved the problem of maintaining the loyalty of subject peoples by offering military protection and civil services in return for taxation, with minimal interference in local customs.