Playing mostly as an inside forward or centre forward (and occasionally as a right-winger), Radford spent most of his career at Arsenal. He joined the club as an apprentice in 1962, turning professional in February 1964. He was a prolific goalscorer in the youth and reserve teams, before making his first-team debut against West Ham United on 21 March 1964 (his only appearance of the 1964-65 season. Radford was used slightly more the next season, playing 15 times, and became Arsenal's youngest ever hat-trick scorer, against Wolves on 2 January 1965, at the age of 17 years and 315 days, a record that remains to this day.
By the start of 1965-66 Radford was an Arsenal regular, and particularly blossomed under the stewardship of Bertie Mee; in 1968-69, although he had been moved out to the right wing, he scored nineteen goals and reached the 1969 League Cup final. As Radford peaked, so did Arsenal; in 1969-70 he again scored nineteen goals, and helped Arsenal win the 1970 Inter-Cities Fairs Cup, their first trophy in seventeen years; Radford scored the second goal in Arsenal's 3-0 win in the second leg of the final, which they won 4-3 on aggregate.
By now, Radford had been moved up front again and continued to score regularly. The following season (1970-71) Radford scored 21, his best single tally in a season, forming a partnership with Ray Kennedy that between them recorded 47 goals. With his goals, Radford was an instrumental part of Arsenal's FA Cup and League Championship double-winning side, and his assists played an important role too; Radford set up Kennedy for the winning goal in Arsenal's FA Cup semi-final replay win against Stoke City, and set up both Eddie Kelly and Charlie George for their goals in the Final against Liverpool.
By now, Radford was an England international, having already won four caps for the under-23 side. He made his full England debut in a friendly against Romania on 15 January 1969. However, he was not a favourite of England manager Sir Alf Ramsey and won only one further cap, against Switzerland on 13 October 1971; Radford scored in neither match.
He continued to play for Arsenal through the early 1970s, scoring another 19 goals in 1972-73. However, his goal rate gradually reduced (only achieving single figures in 1973-74 and 1974-75) and he was injured in 1975-76, further restricting his appearances. By now, the partnership of Malcolm Macdonald and Frank Stapleton had become Arsenal's first-choice attacking duo and Radford only played twice in the first four months of 1976-77. Unable to stake a place in the side, Radford moved on to West Ham United in December 1976 for £80,000. In all he played 481 times for Arsenal, scoring 149 goals, which makes him (as of 2006) Arsenal's fourth all-time top scorer.
After a year and 28 league appearances and no goals with the Hammers, Radford joined Blackburn Rovers in 1977. He was moderately successful with the Second Division side, scoring ten times in 38 league appearances. He left Rovers in 1978 and played for non-league Bishop's Stortford before retiring. After retiring, he became a pub landlord, and enjoyed several spells as manager of Bishop's Stortford in the late 1980s and early 1990s. He now also gives tours of Arsenal's Emirates Stadium and appears as a football pundit on their TV station: Arsenal TV.