Alexandre da Costa was born in Montreal, Canada, in 1979. He was a child prodigy, giving his first public concerts at nine on both violin and piano. In 1998 he earned a master's degree in violin at the Quebec Conservatory, where he also captured first prize in violin performance. That same year he received a bachelor's degree in piano from the University of Montreal.
Da Costa then traveled to Europe and concertized as both a violinist and pianist. His debut CD was issued on the Amberola label in 1998, and featured da Costa performing piano works by Schumann and Pierre Max Dubois, and violin works by Saint-Säens, Sarasate, and Vittorio Monti. Da Costa also continued his education, studying violin from 1998-2001 at the Madrid-based Escuela Superior de Música Reina Sofia with Zakhar Bron, whom he considers his mentor. Da Costa captured first prize at the 1999 Sarasate International Competition. In 2002 the Disques XXI-21 label issued his recording, Hendrix, McCartney & Ysaye, wherein da Costa again performed on both violin and piano, this time in a mixture of rock, jazz, and classical styles. That same year the Canada Council for the Arts presented da Costa with the prestigious Sylva Gelber Foundation Award.
In 2003, after spending five years in Europe, da Costa decided to split his time between Canada and Europe. He won the Musical Instrument Bank Competition that year and was awarded by the Canada Council for The Arts use of the 1689 Baumgartner Stradivarius violin for three years.
Da Costa's 2006 Disques XXI-21 CD of concertos by Portuguese composers Luis Freitas Branco and Joly Braga Santos received a Juno Award nomination. In 2009 da Costa co-founded the chamber orchestra Ensemble Camerata Orford, with whom he has performed at the Orford Festival. In 2010 the Canada Council for The Arts bestowed da Costa with another honor when he was named recipient of the Virginia Parker Prize. Da Costa drew international praise for his 2011 Warner Classics recording of Fire and Blood and other works by Michael Daugherty.