Chuykov studied at the Turkestan School of Art in Tashkent and subsequently, under Robert Falk, in Moscow. He was a People's Artist of the USSR and the winner of the Stalin Prize in 1949 and 1951. He taught in Moscow at the Academy of Arts from 1930 to 1932 and the V. I. Surikov Art Institute in 1947 and 1948. He was awarded a gold medal at the Exposition Universelle in 1959 and his painting, Daughter of Soviet Kyrgyzia, showing a schoolgirl carrying her books, appeared on a Soviet stamp.
Using the traditions of Russian realistic art, Chuykov developed Kirghiz national themes. His genre and landscape paintings depict the life of the people and the natural features of Kirghizia. In the 1950s and 1960s, after travels to India, Chuykov included images from that country in his works. His paintings are noted for spatial clarity, compositional strictness and simplicity, and subtle yet expressive colour relationships. He conveys the significance of the common man in his small-figured compositions, resolving differently in each work the relationship of the figure to the landscape and to the plane of the canvas. As Chuykov's style developed, traits of monumentality appeared increasingly. In his last years he frequently used the triptych form.