He was born in County Wexford. He attended a local Christian Brothers school. He is married to Mary Duggan, an architect, and has three children.

He has been involved with the Wexford Swimming Club and the Wexford Arts Centre.

Fleming attended University College Dublin where he studied for four years. He earned a Bachelor of Science in Experimental Physics. In 1980 he achieved a Master of Science following two years of research in the Atmospherics Group of the UCD Physics Department. When he left university, he moved straight to Irish Met Service. After sixteen months training, he was deployed to Dublin Airport as an aviation forecaster. He moved to the Glasnevin base two years later.

He gave his first weather forecast in July 1985. In 1990, he was made co-ordinator of weather at RTÉ Television. While on television he was noted for his tendency to wink.

He has served as Chairperson of the International Association of Broadcast Meteorology and Co-Chair of the First World Conference on Broadcast Meteorology. He has also chaired the Expert Team on Media Issues for the World Meteorological Organisation.

Fleming is currently Head of the General Forecasting Division in Met Éireann. In this position he has been a representative of Met Éireann to the media. He is a regular contributor to Mooney on RTÉ Radio 1.