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Greenwood was born in Scarborough, Ontario, Canada, and it was clear from an early age she wanted to be a performer; by the time she was in high school at Agincourt Collegiate Institute, she was usually the star of the school plays and concerts. When it came time to graduate in 1980, she feared that she would never get to act again, until a concerned friend gave her a brochure for the American Academy of Dramatic Arts, where Greenwood spent the next two years studying acting.
In 1987, when she returned to Canada in Toronto, she managed to find work for a small night club act called A Wedge of Night, which led to her five-year stint with the Toronto branch of the famous Second City comedy troupe; where she served as both a writer and a performer - two of her shows won Dora Mavor Moore awards in 1988, and 1992. Shortly afterwards, she and her friends Jonathan Wilson and Ed Sahely developed what she called their "Second City" spin-off, Not to Be Repeated, in which the three would improvise an entire situation comedy in front of a live audience, based solely on ideas and suggestions that are thrown at them from the audience. Other notable appearances in Canada include the popular sketch shows This Hour Has 22 Minutes and Royal Canadian Air Farce.
Greenwood is married to television writer John Dolan, with whom she has two daughters, Josephine and Phoebe. Greenwood is also a part of the sketch comedy troupe Women Fully Clothed, featuring what Eugene Levy calls "the five funniest women in Canada." The group saw great success in Canada and even found international acclaim with their appearance in Edinburgh, Scotland.