Plunket was born at Kilsaran, near Castlebellingha, in County Louth. The eldest of 6 children, one of whom died in infancy, she was a granddaughter of William Plunket, 1st Baron Plunket, Lord Chancellor of Ireland. Her father Thomas Plunket, 2nd Baron Plunket (1792–1866), was a Junior Church of Ireland clergyman when she was born and later became the Bishop of Tuam, Killala and Achonry. Her mother Louise Jane Foster (married in 1819) was the daughter of John William Foster of Fanevalley, County Louth, Member of Parliament for Dunleer, and was related to the Earl of Clermont. Her first and second cousins included 3 titled members of the Irish aristocracy. She was baptised Anglican in Kilsaran Church on 13 December 1820 as Catherine Plunket, though she spelled her name with a K for her entire life.

She inherited from her mother one of the family’s ancestral homes, Ballymascanlon House near Dundalk, and oversaw the upkeep of the home and gardens until she contracted bronchitis at the age of 102 (her only serious health problem). The house is now a hotel.

With her younger sister Gertrude (1841–1924), Plunket traveled widely and visited almost every capital in Europe with her sister Frederica (1838–1886) she made many sketches of flowers in France, Italy, Spain and Germany, and Ireland. These were bound in a volume which was presented in 1903 to the Royal College of Sicence, and was later transferred to the Museum of Science and Art in the National Museum of Ireland.