Johnston was born in San José, Uruguay. His father, Fowell Buxton Johnston (born 1839) was an officer in the 3rd Dragoon Guards, and the younger son of Scottish MP Andrew Johnston and his second wife, abolitionist Priscilla Buxton, daughter of Sir Thomas Fowell Buxton, 1st Baronet. Johnston's uncle (his father's elder brother), also Andrew Johnston, became an MP in Essex in the 1860s.
The family returned to England in 1875. With his father seeking work, and his mother ill, Johnston was raised by an aunt. He was educated at home, and enjoyed mathematics, technology, and creating illuminated manuscripts. His mother died in 1891, and he began to work for an uncle. He spent some time studying medicine at Edinburgh University but did not complete the course.
After his mother's death, his father was remarried, to a sister of Robert Chalmers, 1st Baron Chalmers. Johnston's half-brother, Andrew Johnston (1897-1917), was killed when his aeroplane crashed while serving in the Royal Flying Corps in the First World War.
After studying published copies of manuscripts by architect William Harrison Cowlishaw, and a handbook by Edward F. Strange, he was introduced to Cowlishaw in 1898 and then to William Lethaby, principal of the Central School of Arts and Crafts. Lethaby advised him to study manuscripts at the British Museum, which encouraged Johnston to make his letters using a broad edged pen. Lethaby also engaged Johnston to teach lettering, and he started teaching at the Central School in Southampton Row, London, in September 1899, where he influenced the typeface designer and sculptor Eric Gill. From 1901 he also taught a class at the Royal College of Art and many students were inspired by his teachings.
He published a handbook, Writing & Illuminating, & Lettering in 1906. He started a second book in the 1920s but it was unfinished at his death. In 1913, Frank Pick commissioned him to design a typeface for London Underground, and the simple and clear sans-serif Johnston typeface was the result.
He has also been credited for reviving the art of modern penmanship and lettering single-handedly through his books and teachings. Johnston also devised the simply crafted round calligraphic handwriting style, written with a broad pen, known today as the foundational hand (what Johnston originally called a slanted pen hand, which was developed from Roman and half-uncial forms). He was influential on a generation of British typographers and calligraphers, including Graily Hewitt, Irene Wellington, Harold Curwen and Stanley Morison, Alfred Fairbank and Eric Gill. He also influenced the transition from Gothic to Roman letters in Germany, and Anna Simons was a student. He also lectured in Dresden in 1912. In 1921, students of Johnston founded the Society of Scribes & Illuminators (SSI), probably the world's foremost calligraphy society.
He met Greta Grieg, a Scottish schoolmistress, in 1900, and they were married in 1903. They had three daughters. They lived in London until moving to Ditchling, Sussex in 1912, where Eric Gill has settled in 1907. His wife died in 1936. He was appointed a CBE in 1939. He died at home in Ditchling.