Born in Tiflis, Aleksi-Meskhishvili left the city at a young age to study medicine. However, after completing his studies, he chose the path of a teacher instead of a doctor when he returned to the capital.
Soon, he obtained the position of director of the Tiflis Dramatic Theatre (1890-1896). Later, he moved to Kutaisi, where he led the Kutaisi Dramatic Theatre from 1897 to 1906.
In 1906-1907, Aleksi-Meskhishvili performed at the Moscow Art Theatre, after which he returned to direct the Tiflis Dramatic Theatre again from 1910 to 1914. From 1916 to 1918, he worked in Russia, where he appeared in several early films.
In 1930, Aleksi-Meskhishvili was posthumously awarded the title of People’s Artist of the Georgian SSR, and ten years later, the Kutaisi Dramatic Theatre was named in his honor.