Finkelstein studied photography under Alexey Brodovitch, the art director of Harper's Bazaar and worked as a photojournalist for the Black Star and PIX photo agencies, reporting primarily on the political developments of various subcultures in New York City in the 1960s.
Finkelstein graduated from Stuyvesant High School in 1950 and in 1952 he enrolled in Brooklyn College, where he first became interested in photography through the inspiration that he found in great photographers such as Edward Steichen.
It was also here that developed his militant political tendencies, to the extent that he was expelled during his final semester after he threw a filing cabinet through a window to protest censorship of a campus publication.
After his expulsion, he managed to acquire an internship with the art director of Harper's Bazaar, Alexey Brodovitch (who famously brought Cocteau, Chagall, and Man Ray to illustrate the magazine).
Finkelstein was signed by the PIX and Black Star agencies (the latter supplied Life magazine with much imagery) through which he was able to meet and spend time with established photographers Robert Capa, Eugene Smith, and Andreas Feininger.
He specialized in chronicling the various subcultures of the United States at the time, an interest that led him to Harlem's burgeoning jazz and soul scenes, Warhol's factory, and later to cover the antiwar rallies and emerging counterculture.
His iconic images of the include subjects such as the Velvet Underground performing live, Marcel Duchamp, Bob Dylan, Edie Sedgwick, Salvador Dalí, and Allen Ginsberg.