What is the background for Cicero's exile from Rome?
This is a great question. If you want the background information on Cicero's exile, then you need to start with the conflict between Clodius and Cicero.
Cicero testified against Clodius, a notorious figure in the late Republic, when Clodius broke into a religious festival reserved only for women (the festival of the good Godess). The Romans took things like this seriously and Clodius went on trial. To make a long story short, Clodius was acquitted, but because Cicero testified against him, a rivalry started. They became "sworn enemies."
Clodius took advantage of his acquittal and got himself elected as a tribune of the plebs by getting himself adopted by a plebian. One of the first things he did was to pass a law against executing a Roman citizen without a trial. He was obviously aiming at Cicero and at his perceived greatness in the Catilinarian conspiracy - where Cicero stopped a potential usurpation of the Republic by Catiline. In this way, Clodius was going after what Cicero viewed as his crowning achievement.
Clodius was able to get a law passed (lex de exsilio Ciceronis) through political maneuvering and Cicero was forced into exile to Greece.