I'm sure that Fogg knows how he made his own fortune (or inherited it); however, Fogg, the narrator, or a different character never explains to readers exactly how he became so wealthy. Fogg is a member of the British aristocracy. He's a wealthy gentleman that appears to have nothing but time on his hands and things to spend money on. That is the general appearance and stereotype of this class, and audiences are often given the impression that someone like Fogg is wealthy and always has been wealthy. He could very well have been born into the money. Often, this kind of thing creates a character that is not likable or relatable; however, that is not the case with Fogg. He doesn't flaunt his wealth by buying and doing frivolous things. He's fairly humble in general, and he's humble about his wealth. That said, he's not afraid to use large sums of money for a cause or a purpose. That is especially true when it comes to helping out a friend such as Passepartout.

We're never told exactly how Phileas Fogg came to be so incredibly rich. All that we know for sure is that he's a very wealthy man, albeit one who leads a fairly modest lifestyle. There's certainly nothing ostentatious or flashy about Fogg. He doesn't need to show off his vast wealth, but he does enjoy its trappings, nonetheless. Fogg's not very forthcoming about the provenance of his riches, and this merely adds to the air of mystery about the man. Even his closest friends and acquaintances haven't the faintest idea how he came by his money. But he genuinely is a rich man, all the same. And the evidence for this is extensive. He lives in a mansion on the fashionable Saville Row; he dines on the finest food whenever he drops by at the Reform Club; he has a line of credit at the world-famous Baring's Bank; and to top it all, he has a full-time manservant, none other than his faithful travelling companion, Passepartout.

Was Fogg rich?

Yes, Phileas Fogg was rich.  

Was Phileas Fogg rich? Undoubtedly. But those who knew him best could not imagine how he had made his fortune, and Mr. Fogg was the last person to whom to apply for the information.

The above quote comes from the first chapter.  What's interesting about Fogg and his wealth is that nobody knows where the wealth comes from.  The paragraphs preceding the above quote explain that Fogg was never seen in the banking district, or at the docks, or at an office, or the courts.  He's not a scientist either, so it is mystery where his wealth comes from.  But there is no doubt that he is wealthy.  It is known that he has an open credit line with the Barings and that "his cheques were regularly paid" from "flush accounts."  

Additionally, he lives in a mansion on Saville Row, and Fogg only dines on the best that the club can offer.  

When he breakfasted or dined all the resources of the club—its kitchens and pantries, its buttery and dairy—aided to crowd his table with their most succulent stores; he was served by the gravest waiters, in dress coats, and shoes with swan-skin soles, who proffered the viands in special porcelain, and on the finest linen; club decanters, of a lost mould, contained his sherry, his port, and his cinnamon-spiced claret; while his beverages were refreshingly cooled with ice, brought at great cost from the American lakes.

Fogg also has a full time servant.  That's a privilege for the wealthy.  Lastly, the reader knows that he is wealthy because he casually bets 20,000 pounds that he can circumnavigate the globe in 80 days or less.  

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