Gaius Caesar's father was an upper class person of good origin although the material wealth was quite small. He was a minor state official. He was a member of the Julia Clan whom are descendants of Iulus, son of Lenas, mythical creator of Rome His mother rose from a less important towards one of greater power.
Caesar and Sulla
Cornelus Sulla who lived from 138-78BC was elected consul in 88BC. He launched a counter strike in Asia after 80,000 Roman soldiers were massacred and in 83 BC came back victorious. With his army he became more powerful so as to become dictator. During this time he created public lists of his enemies to be killed and at least a thousand perished due to this.
Caesar was no longer welcome in Rome due to is connectiongs Marius and Cinna. He lost the inheritance from his father and lost the dowry of his marriage to Conclia. He was no longer a priest and was given freedom from harrasment but went to Asia to be an aide to Marcus Thermus.
He won the citizen's crown after having stormed Mytilene on the Island Of Lesbos. Following from this his reputation had increased and it would become time for him to rise to power.
For further information why not check out Julius Caesar: A beginner's Guide by Antony Kamm?