Connelly was born in Calgary, Alberta. At seventeen, she lived in a Thai village thanks to a Rotary exchange scholarship. She returned to Canada a year later. At nineteen, she left for Spain, where she lived almost two years. Having no work visa, she supported herself by, among other things, teaching English as a second language. In her spare time, she wrote about her experiences and took photographs with which to illustrate her writing. She also reworked the letters and journals, which she had written in Thailand, into a manuscript that was to become Touch the Dragon.

In 1991, she moved to France and settled in Montclar, Avignon, where she studied French and Spanish. Soon after, she travelled to Greece, spending most of her time on the island of Lesbos, to which she has occasionally returned. She then moved back to Canada for an extended period, writing and promoting her work.

Her first book, a poetry collection entitled The Small Words in My Body (1990), won the Pat Lowther Award for poetry in 1991. Her second book, Touch the Dragon: A Thai Journal (1992), won the Governor General's Award for non-fiction.

Three poetry collections followed, This Brighter Prison (1993) The Disorder of Love (1997) and The Border Surrounds Us (2000). She also compiled a book of letters, One Room in a Castle, detailing her experiences in Europe.

In 1996, she returned to Thailand and also went to Myanmar (she prefers the older name, Burma) seeking material for another book. Her experiences there served as the basis for her novel about a political prisoner, The Lizard Cage, which won the Orange Broadband Prize for New Writers and was longlisted for the 2007 IMPAC Award.

She stayed in Thailand for two years before returning to Canada, where she married. She lives in Toronto.