He did not captured Rome. He sacked it and then went to Calabria
(the toe of Italy) and died there. Had he stayed in Rome he would
have had to face the coming of the Roman legions gathered from
around the empire. Moreover, he did not want to undermine the
empire. He was acting out of his grievance for not having been
appointed as a general of the Roman army as he had expected.
This grievance led him to attack northern Italy, but was
defeated by Stilicho, the chief of staff of the army of the west.
The treacherous murder of Stilicho made the situation bad. Stilicho
was the only commander who was capable to defeat Alaric. Without
him Italy became vulnerable. Stilicho also had made an alliance
with Alaric. They planned to attack the part Illyrucum which
belonged to the empire of the east together. This would have
satisfied Alaric's ambitions.
The young and incompetent emperor Honorious mishandled Alaric's
demand of a large sum of money as compensation for the cancellation
of the campaign in Illlyrucum which followed Stilicho's death. As a
result, Alaric besieged Rome three times. The third time he sacked
it.
It is likely that Alaric went to Calabria because he planned to
go to Africa (the breadbasket of the empire) to disrupt Rome's
grain supplies.
