It was hardly uncommon for politically-inclined Romans to enter into legal advocacy, particularly early in their careers. Advocacy allowed you to display your skill as an orator, a crucial part of political life, and to build up relationships with important people throughout the political scene--the concept of mutual reciprocity was a very important one among the upper strata of Roman political society. You did a favor for somebody, so now he owes you one, a favor which was often repaid in votes and support while canvassing for office. Cicero is the best example of this, since he secured his election to the quaestorship, despite being a novus homo, but acting as a defense attorney--he notoriously defended literally anyone who asked him, and so got a huge number of votes when he stood for quaestor. This was an important stepping-stone, establishing political relationships and rhetorical competence before embarking on lower magistracies like the military tribunate and eventually starting the cursus honorum. Neither Cicero nor Caesar are particularly unusual in this respect.

Plutarch gives a record of two of Caesar's early cases, his prosecution of Dolabella in 77 and his representation of the free Greek cities against Publius Antonius:

ἐπανελθὼν δ᾽ εἰς Ῥώμην Δολοβέλλαν ἔκρινε κακώσεως ἐπαρχίας, καὶ πολλαὶ τῶν ἀπὸ τῆς Ἑλλάδος πόλεων μαρτυρίας αὐτῷ παρέσχον, ὁ μὲν οὖν Δολοβέλλας ἀπέφυγε τὴν δίκην, ὁ δὲ Καῖσαρ ἀμειβόμενος τὴν Ἑλλάδα τῆς προθυμίας συνηγόρευσεν αὐτῇ Πόπλιον Ἀντώνιον διωκούσῃ δωροδοκίας ἐπὶ Λευκούλλου τοῦ Μάρκου Μακεδονίας στρατηγοῦ, καὶ τοσοῦτον ἴσχυσεν ὥστε τὸν Ἀντώνιον ἐπικαλέσασθαι τοὺς δημάρχους, σκηψάμενον οὐκ ἔχειν τὸ ἴσον ἐν τῇ Ἑλλάδι πρὸς Ἕλληνας,

But having returned to Rome he prosecuted Dolabella for ill-treatment in the governing of his province, and many of the cities of Greece furnished him with their testimony. While in fact Dolabella escaped punishment [that is, he was acquitted], in return for their good-will [προθυμία is kind of a hard word to translate. It means "ready kindness, zeal, goodwill"--the sort of favor done out of the goodness of one's heart, not necessarily with the expectation, as might be excepted in Roman society, of a favor in return], served as their advocate when they prosecuted Publius Antonius, for taking bribes, before Marcus Lucullus, the governor [literally "general"--Lucullus was praetor at the time] of Macedonia. And he was so successful that Antonius appealed to the tribunes, alleging that he could not have a fair trial in Greece against Greeks

Plutarch notes that these trials, in their support of the common people, were indicative of Caesar's early advocacy:

ἐν δὲ Ῥώμῃ πολλὴ μὲν ἐπὶ τῷ λόγῳ περὶ τὰς συνηγορίας αὐτοῦ χάρις ἐξέλαμπε, πολλὴ δὲ τῆς περὶ τὰς δεξιώσεις καὶ ὁμιλίας φιλοφροσύνης εὔνοια παρὰ τῶν δημοτῶν ἀπήντα, θεραπευτικοῦ παρ᾽ ἡλικίαν ὄντος.

But indeed at Rome he won great favor by his eloquence in advocacy, and met with much goodwill from the people for his greeting them and his friendly intercourse with them, for he was seeking office beyond his years.

This resembles Cicero's later prosecution of Verres on behalf of the province of Sicily, although unlike Caesar Cicero had always made it something of a doctrine to never represent the prosecution, which he first broke against Verres. Caesar jumped right in with prosecution, and it appears generally with cases representing the greater public. Badian suggested that this might indicate that from an early age Caesar was concerned with the state of the public, pointing to Caesar's close relationships with the people, even the scum of the city, during much of his early life, particularly while he was on the run from Sulla and during his time living in the Subura. It's an interesting thought.

Suetonius also mentions the prosecution of Dolabella, saying that Caesar fled to Rhodes, because he had stirred up a lot of the senatorial class against him and made himself rather unpopular with them--it was then that he studied with Apollonius.