When Antoine Bechamp discovered his reduction, it was through reducing nitronaphtaline and nitrobenzene with iron and acid. When nitrobenzene (C6H5NO2) is reduced, you get aniline (C6H5NH2)---where the NO2 group on the aromatic has been reduced to NH2. This reaction did not produce mauveine (C26H25NH4 or C27H25N4).

Two years later, Perkin was trying to synthesize quinine, the natural principle of the cinchona tree used as a therapeutic for fevers. The proposed process was to oxidate 2 parts N-allyl toluidine (C10H13N) into one quinine (C20H24N2O2). This did not work, the reaction produced a dark colored mess. However, upon cleaning his reaction vessel, Perkins noticed the purple color that came off the black precipitate when washed with ethanol. The reaction that produced this new material, as was determined latter, had occurred among the mixture of impurities in the N-allyl toluidine---aniline, p-toluidine, and o-toluidine: the constituent parts of the Perkin's mauve (C26H25NH4 or C27H25N4).

As Anthony S. Travis writes in his article on the subject, the young Perkin was discouraged from pursuing this product because aniline was relatively scarce, making the dye recipe an expensive prospect. The problem here went beyond just the Bechamp reaction, including the entire "pipeline" of hydrocarbons that could be extracted from coal tar. The Bechamp process was just one step in the pipeline, a step that did not itself produce the dye (again, Bechamp made aniline, not the whole mauve molecule). However, Perkin picked a remarkable time to pursue this line of work, as French and English commercial chemical operations were winding up and new processes (like Bechamp's) were being put into practice on a larger scale---where nitrobenzene was being produced from benzene, benzene was acquired through fractioning from coal tar, etc down the line. The effect of this would be that in the convergence of different industrial and scientific advances, Perkin's mauve would become a major product of the incipient chemical industry.

EDIT: I don't know why I can't get subscripts to work, sorry about that.