Dodgin was born in Wardley, Gateshead, and is the son of footballer Bill Dodgin Sr. He began his career as an amateur, before signing for Southampton, whom his father managed. When Dodgin Sr. moved to Fulham in 1949, his son followed him there, although he still had to wait two years before making his debut in December 1951, against Preston North End at left back. He soon switched to right back, but could not prevent Fulham from being relegated to the Second Division. Feeling the pressure from the fans, Dodgin transferred to Arsenal for a fee of £4,000 in December 1952, having played 35 League matches for the Cottagers. By now, he was a centre half, and after a spell in the reserves he made his debut against Bolton Wanderers on 15 April 1953. Arsenal won the old First Division in 1952–53 but the Bolton game was his only contribution to that success. After the departure of regular centre half Ray Daniel to Sunderland that summer, Dodgin became first choice at the back for Arsenal, missing only three matches that season. However, despite his height Dodgin was a rather cumbersome defender, and could not live up to the performances of his predecessor. He was dropped at the start of 1954–55 in favour of Jim Fotheringham. He returned in 1956–57 and was a regular in the side for the next four seasons. All through this time, Arsenal were going through a barren patch, and apart from a third-place finish in 1958–59, they were never close to winning a trophy. In total he played 208 matches for Arsenal, scoring one goal. Dodgin was given a free transfer in March 1961 and returned to his old club Fulham, helping them to an FA Cup semi-final the next season. However, he broke his leg in a match against Aston Villa in 1962 and the injury effectively ended his career; he played only seven more games, his last game coming against his other old club, Arsenal.
International career
Although Dodgin played for and captained England at under-23 level, he never appeared for the full international team.
Management career
After retiring from the game, Dodgin became a coach first at Millwall, then at Queens Park Rangers, where he helped the club to its 1967League Cup win. He became QPR's caretaker manager in August 1968 before leaving the club in November of that year. During his tenure at Loftus Road, he presided over the club's worst start in its history which was a run of 12 games without a win. The club thereafter got its first league win of the season by beating fellow promotees Ipswich Town 2–1 in the 13th match of the season. He then took the manager's job at Fulham in December 1968. Although the club were relegated from the First Division to the Third in successive seasons, Dodgin stayed with the club and led them to promotion to the Second Division in 1971. However, Fulham struggled to stay afloat in the Second Division, and despite avoiding relegation in 1972 he resigned. Dodgin later managed Northampton Town and Brentford, winning promotion from the Fourth Division with both clubs. He then managed Northampton Town for a second, less successful spell and finally Woking before retiring. He died after a long battle with Alzheimers in 2000, aged 68.