How does the original meaning of racket lead to the following metaphoric meanings?
an illegal or dishonest scheme for obtaining money
a person’s line of business or way of life
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Racket has had since the 17th century two broad and probably unrelated senses:
The implement employed for striking a ball or shuttlecock in games like tennis, and by extension such games themselves—particularly "rackets", which may have evolved from tennis in the London debtors' prisons.
Noise or tumult; this sense was extended in the 18th century to noisy social gatherings and to disorderly conduct in general.
About the turn of the 19th century these two terms seem to have been conflated in English underworld slang—the flash language of the family—as a synonym for game in the sense of one or another particular method of criminal endeavour. The earliest use I have found (and OED 1 concurs) is in A New and Comprehensive Vocabulary of the Flash Language, compiled and written by James Hardy Vaux. Written 1812, and appended to Volume I of his Memoirs, 1819.
RACKET, some particular kinds of fraud and robbery are so termed, when called by their flash titles, and others Rig; as, the Letter-racket, the Order-racket; the Kid-rig; the Cat and Kitten-rig, &c., but all these terms depend upon the fancy of the speaker. In fact, any game may be termed a rig, racket, suit, slum, &c., by prefixing thereto the particular branch of depredation or fraud in question, many examples of which occur in this work.
[I list some related terms from this fascinating work below; the book itself is available online here.]
ADDED: It is possible that a third sense of racket contributed. Racket is found in some 18th-and 19th-century texts as a variant of ratchet, a toothed wheel or bar, or a tooth or click for stopping the action of such a wheel or bar, and the general notion of device might have fed into the meaning.
This sense survives in modern rackets, racketeering. The jocular extension of the term from one's particular criminal specialization to one's particular non-criminal specialization is unremarkable.
CROSS, illegal or dishonest practices in general are called the cross, in opposition to the square. See SQUARE. Any article which has been irregularly obtained, is said to have been got upon the cross, and is emphatically termed a cross article.
FAMILY, thieves, sharpers and all others who get their living upon the cross, are comprehended under the title of “The Family.”
FLASH, the cant language used by the family. To speak good flash is to be well versed in cant terms.
FLASH, to be flash to any matter or meaning, is to understand or comprehend it …
GAME, every particular branch of depredation practiced by the family, is called a game; as, what game do you go upon? One species of robbery or fraud is said to be a good game, another a queer game, &c.