Poore was born in Paddington, London. His father, Roger Poore DSO, was killed in action during the First World War on 26 September 1917. Poore married Peta Farley née Coast on 24 March 1949. They had one daughter, Victoria Borwick, who later became a Member of Parliament for Kensington.
Poore's primary early discipline was hill climbing. In 1950 he drove a 3.8-litre twin-Wade-blown Alfa Romeo to win the British Hill Climb Championship. His campaign included a second place at Shelsley Walsh, first at Prescott, second at Bo'ness, first at Rest and Be Thankful, second at Bouley Bay, first at Val des Terres, and a final victory at Prescott to clinch the title. It was a comprehensive points campaign across the breadth of British hill climb venues.
Poore made two appearances in the Formula One World Championship in 1952, both for Connaught Engineering in a Connaught Type A powered by a Lea-Francis four-cylinder engine. His Connaught teammates across those two rounds included Ken McAlpine and Stirling Moss at the British race, and McAlpine again at the Italian.
At the 1952 British Grand Prix on 19 July, Poore started from eighth on the grid and finished fourth, scoring three championship points — a strong result in a field that included the dominant Ferrari entries of the era. He finished twelfth at the 1952 Italian Grand Prix driving for the Connaught Racing Syndicate. He ended the 1952 season thirteenth in the World Drivers' Championship, his only classified season.
From the early 1950s Poore raced sportscars for Aston Martin and the associated Lagonda marque. He made four consecutive entries at the 24 Hours of Le Mans, none of which reached the finish: an Aston Martin DB3 in 1952 (DNF), an Aston Martin DB3S in 1953 (DNF), a Lagonda DP115 in 1954 (DNF), and a Lagonda DP166 in 1955 (DNF). His most significant sportscar result came not at Le Mans but at the 1955 Goodwood International Nine Hours, which he won co-driving an Aston Martin DB3S with Peter Walker.
Poore used his personal wealth to bankroll the founding of Autosport, the weekly motor racing journal, in 1950 — the same year he won the hill climb championship. The magazine became the leading English-language weekly dedicated to motorsport across the following decades.
Poore became chairman of Manganese Bronze Holdings, an engineering conglomerate whose primary business was marine propellers. He sold that core business and used the proceeds to acquire a group of failing British motorcycle manufacturers: Associated Motor Cycles (which brought AJS, Matchless, James, and Francis-Barnett), Norton, and engine maker Villiers.
Following the collapse of BSA Group in 1972, the motorcycle interests of Manganese Bronze and those of BSA and Triumph were consolidated into Norton Villiers Triumph (NVT), with Poore as chairman. As NVT he made 3,000 of the 4,500-strong combined workforce redundant, a restructuring that triggered the formation of the Meriden Workers' Cooperative — which operated the Triumph Meriden factory independently for approximately ten years before folding. Production of the BSA A65 twin and A75 Rocket Three triple ceased entirely; Triumph was absorbed by the Cooperative. The sole NVT model to survive was the Norton Commando, which won the Motor Cycle News Machine of the Year award for five consecutive years.
NVT ultimately ceased production and later assembled an Asian-made 125cc trail bike under the Norton name. The firm subsequently developed a twin-rotor Wankel-engined motorcycle based on David Garside's earlier BSA research.
The BSA acquisition also brought Carbodies, the manufacturer of the FX4 London taxi — the vehicle widely known as the black cab. After the motorcycle interests were wound down, Poore continued to chair Manganese Bronze in its role as a taxi and engineering component manufacturer until his death in February 1987.