Alexander Johnson studied piano with Joseph Stanford and composition with Henk Temmingh at the University of Pretoria, graduating in 1990 with the degree BMus (Performing Arts) (cum laude). In the same year he also obtained the Performer's Licentiate in Music (Piano) (cum laude) from the University of South Africa. He continued his postgraduate studies at the University of Pretoria, obtaining the BMus (Hons)(Performing Arts) (cum laude) in 1991, an MMus in Composition (cum laude) in 1999, and a DMus in Composition in 2000.
Johnson made his concert debut as a pianist in 1990, playing Ravel's Concerto for the Left Hand with the Natal Philharmonic Orchestra. This was broadcast live in South Africa. He has since performed as soloist with various orchestras in all the major concert venues in South Africa. Johnson has also made several recordings as a pianist for the South African Broadcasting Corporation, both for radio and television.
After having being awarded two national composition prizes from the Foundation for the Creative Arts consecutively in 1993 and 1994, Johnson moved to Belgrade, Serbia, where he composed his first Piano Concerto, Nish. This was premiered in Nish, Serbia, by the Croatian pianist Dorian Leljak, with the Nish Philharmonic Orchestra conducted by Anatoli Novitski. At this time, Johnson's Sonatina for Clarinet and Piano was performed at the International Festival of Contemporary Music at the SAVA Centre in Belgrade.
On his return to South Africa in 1994, Johnson composed Monyanya, a Concerto for two pianos, violin, chorus and orchestra, a work commissioned by the Delegation of the European Union in South Africa for the celebration of the first Europe Day in South Africa on 9 May 1995. In 1994, Johnson's Jazz Impromptu was the prescribed composition at the Seventh International UNISA-Transnet Piano Competition.
His Sonatina for Clarinet and Piano has been performed throughout Europe and South Africa by the French clarinetist Gilles Swierc.
In 1999, the University of Pretoria granted Johnson a scholarship to research his ballet, Thunderbird, based on a native Canadian myth. This was first performed in 2000 in Pretoria. His Suite South Africa for violin and string orchestra was given its world premiere by the Lucerne Festival Strings in Pretoria in 2002. In 2010 his Afro-disiacs for String Orchestra was performed by the Zürich Camerata Orchestra conducted by Marc Kissoczy. His Miniature Incantation for Piano represented South Africa in the world's first Commonwealth Pianorama, linked to the Commonwealth People's Forum and the Commonwealth Heads of Government meetings (CHOGM), which took place in Trinidad and Tobago in 2009. It was also performed by Sean Jackson (UK) in London and in Edinburgh. Johnson's newly composed Imicabango for Flute and Piano received its premiere performance by South African flautist Cobus du Toit and pianist Jannie le Roux in September 2010. Smaller works for piano have over the past 20 years regularly appeared as examination pieces in the University of South Africa's syllabi. Johnson was also the co-founder and director of the former Artium Symphony Orchestra. He is currently the head of the Music Department at the University of Pretoria, teaching harmony, counterpoint, form, analysis, composition and orchestration.