Aleksey graduated from the Constantin Institute for Border-lines (Russian: Константиновский Межевой Институт) in Moscow. After graduation, he attended classes at the Moscow University and Sorbonne. In 1860 he became an Instructor of Mathematics at the Constantin Institute. He obtained degrees of Master (1868) and Ph.D. (1874) from the Moscow University. In 1868 he became a Professor of the Imperial Moscow Technical School, in 1879-1880 he was an Inspector of the School. From 1883 he was the principal of the Alexander Commercial School (Russian: Александровское коммерческое училище, currently the State University of Management, ); from 1884 he was a Corresponding Member of Russian Academy of Sciences. Letnikov died in 1888 in Moscow and buried in Alexandroskoe cemetery .
Letnikov's most renowned contribution is the creation of the Grunwald-Letnikov differintegral. He also published results about analytic geometry, ordinary differential equation and non-Euclidean geometry.