Kapranos spent his early childhood in Sunderland and South Shields, his mother's home town, and spent most summers in his father's native Greece. Huntley is the name of his father's aunt. His mother insisted on the middle name Paul, because of a long-term crush on Paul McCartney.

In 1980, he moved with his family to Edinburgh, then to Glasgow in 1984. At the age of seventeen, Kapranos attended the University of Aberdeen to study theology. He continued his studies at several different universities, including the University of Strathclyde in Glasgow. From the early 1990s, he was a fixture of the Glasgow music scene, running live nights at the 13th Note, most notably The Kazoo Club (where bands such as Picnic, and The Oddballs were regulars). While working as a chef, bartender, lecturer in IT at the city's Anniesland College, and other various jobs, he played in some of Glasgow's popular bands, including The Blisters, long-standing ska stalwarts The Amphetameanies, The Yummy Fur, Quinn and The Karelia (Band) (formerly known as The Blisters). He is also known to have contributed to Urusei Yatsura recordings.

Eventually, after dropping "Huntley" from his name, he formed a new band in 2001 with old friends and new acquaintances called Franz Ferdinand, which has received international attention. The band comprises of Alex, Nick McCarthy (guitar, vocals), Paul Thomson (drums) and Bob Hardy (bass). Five singles and an EP have charted in the United Kingdom to date, and their eponymous debut album won the 2004 Mercury Music Prize, two Brit Awards, a Breakthrough Video MTV Award and an Ivor.

Kapranos has asthma and is allergic to animals and peanuts, which he likens to arsenic.

In September 2005 he began Soundbites , a weekly food column for G2 in the Guardian newspaper, which details his culinary adventures as Franz Ferdinand traverse the globe on their world tour.