One of the important distinctions here regards what philosophical school you subscribe to (or are helping to develop), as this would play a big role in how you, "made your living" or went about finding adherents.

Remember that several of the major ancient Greek philosophers eschewed wealth, or at least sought to minimize its hold on their lives, and that played a big role in how they operated.

Take Cynicism, which got its start in Athens in the 5th century BC., but eventually spread through the Roman Empire. Its adherents could be found living and preaching around the Mediterranean during the 1st and 2nd century, and they did it in an extremely distinct style.

Crates of Thebes, one of the heads of the Cynic school, famously gave away a large fortune so he could live out his philosophic ideal life of poverty on the streets of Athens.

Crates begged for food, slept in public buildings, and regularly engaged in teaching/conversation with the public at large.

Fascinatingly, Hipparchia, daughter of a wealthy Thracian family, fell in love with him and gave up her wealthy family to live with Crates in poverty.

Crates was regularly insulted because of his poverty and life on the streets, and called, a "dog." When someone asked him why he was called this, he turned the insult around and used it explain how he operated: "I protect my friends, growl at the untrustworthy, and bite wrong-doers," he told them.

A modern person might find some of his outreach techniques to be disturbing.

He was called, "The Door-Opener" because he would walk into people's houses and lecture them on morality. Yet, people seemed to appreciate him, and apparently families in conflict would call him in so that he would be able to mend family problems - kind of like a therapist.

Sources:

Epictetus, Discourse 3.22, On Cynicism The Apologia and Florida of Apuleius of Madaura, H. E. Butler, translator, Oxford Press (1909) The Cynics: The Cynic Movement in Antiquity and Its Legacy

You’re a young up-and-coming philosopher, or sophist, because you likely don’t have to worry about making a living. You’ve gotten an education only because your family owns wealth-producing estates or, perhaps, are wealthy merchants, but most likely you are from the landed gentry, to use an English expression. Although there is some disagreement, it seems that the majority of common people were not literate, and few people had access to formal education, so you probably do not come from that class. Although a number of philosophers preached (and practiced) a simple life, I think it’s much easier for wealthy people to adopt a philosophy of poverty than it is for people who’ve grown up in poverty and want. Poor people never really get a break from trying to eke out a living, and certainly don’t have the time or money for luxuries like education.

But, once you’ve gotten an education, and perhaps studied under a well-known master, and then have made a name for yourself, you can open a school if you like and take in some money that way. Sophists made their living by teaching and speaking, and particularly by teaching the art of rhetoric. To have a successful public life, a man had to communicate to the people through speech-making, so learning rhetoric and logic was considered very important and valuable. You would have number of wealthy clients available to you.

You could also function as a kind of lawyer, or really, an advocate. People would hire men more eloquent than themselves, to defend themselves, or to bring a case in the Agora where you might have to persuade hundreds of jurors at once. It was these juries who made all decisions in the Agora, so being a persuasive speaker was critical. The Archons, or “judges" performed purely administrative functions, and so Athenian trials were all about swaying the jurors with rhetoric, rather than about, say, fine points of the law.

Another possibility might be to have a wealthy patron. I’m not sure how common this was in Athens, nor am I aware of a specific example, but I do know these kinds of relationships existed in classical Rome, and throughout Europe during the Middle Ages and beyond.

For sources on how philosophers in ancient Athens lived, I suggest you look at Diogenes Laertius' 'Lives of the Philosophers', Plato's early dialogues (especially the Euthydemus, Charmides, Lysis, Gorgias), and Xenophon's Socratic works, particularly the 'Memorabilia'.

The short answer is that most young men engaged in philosophy would have been aristocrats who owned land and slaves to farm it. Socrates was the child of a stonemason and a midwife (this is mentioned multiple times by Plato and Xenophon; Plato's Socrates is especially keen on getting metaphorical juice out of the midwife thing, see the 'Theaetetus'), but that was unusual; Plato was far more typical. He was an aristocrat; his uncle was one of the Thirty Tyrants, who briefly took over Athens in an anti-democratic coup. In his early dialogues we meet a lot of other men interested in philosophy, most of them young, most of them also aristocrats.

If you need to earn your living, your best bet is probably to get rich men to employ you to educate either them or their sons. This is the approach favoured by the sophists in Plato's 'Euthydemus', for example. Essentially you're becoming a private philosophy-and-argumentation teacher. If you become especially famous, you may be able to set up a school, like Plato's Academy or Aristotle's Lyceum, or to get hired to teach a foreign ruler or his son, like Aristotle's tutoring of the young Alexander (cf Diogenes Laertius' entry on Aristotle) or Plato's excursions in Sicily (cf Plato's letters VII and VIII). That's a rare outcome, though - most likely you'll be teaching people, in their own homes or out of doors, for your whole life. If you can offer a practical benefit to such teaching, like improved outcomes in court cases (the Athenians were fairly litigious), you will appeal more to parents; if you have some sort of 'Rockstar' quality or general novelty, you'll appeal more to the young. This is the situation portrayed in Aristophanes' 'Clouds': it's satirical and shouldn't be taken too literally, but the 'Socrates' he portrays is running a wacky school that essentially fleeces young people and their parents, offering both improved ability to argue in court (in this case, to wriggle out of debts) and a trendy sophistication, but at a cost.