Dodds was born and brought up in Dundee and lived close to the city's two football stadia, regularly attending both Dundee United and Dundee fixtures. He started his playing career at Celtic Boys Club, aged 9, where he was played at left-back. However, at the age of 14 he signed schoolboy forms with local club Dundee United. Developing into a promising striker, he turned professional with United in 1975, but was initially part-time while also serving an apprenticeship as a painter and decorator, his father's occupation.
Club career
Dundee United
Dodds signed for Dundee United on an S form from Hillside Boys Club in December 1972. Within a couple of years he was appearing in the reserves, and signed a professional contract with United in 1975. He made his first team debut for United against Arbroath in August 1976, scoring twice in a 3-1 victory, but it took several years for him to fully establish himself as a first team regular. During the 1977–78 season, he played several games for Arbroath on loan. Dundee United at this time were developing into one of the strongest sides in Scotland. Despite playing in some of the earlier rounds, Dodds missed out on the Scottish League Cup final in 1979 when the club lifted their first ever major trophy, but the following year he was part of the side that defeated city rivals Dundee in the final to retain the cup. He scored United's first goal in a 3–0 victory at Dundee's Dens Park ground. By this time Dodds was featuring regularly in the side and forming a profitable attacking partnership, as a 'target man' alongside the diminutive Paul Sturrock. In 1982–83 they were part of the United side that lifted the Scottish League title for the first time, with Dodds contributing 22 goals. The following season saw Dodds star in the club's run to the European Cup semi-finals. He would also finish on the losing side in two Scottish Cup finals with United, in 1981 and 1985. Despite this run of success, Dodds was becoming unsettled at Dundee United. Despite the efforts of manager Jim McLean to keep the player at Tannadice, he left the club at the end of his contract in 1986.
Later career
In summer 1986, Dodds joined Swiss club Neuchâtel Xamax, whom he had played against for Dundee United in the previous season's UEFA Cup campaign. United received a fee of £180,000. However, personal problems led to the player seeking a move back to Scotland within months, and he was brought to Aberdeen by Alex Ferguson, although within weeks the manager who signed him had moved on. Dodds' most significant contribution during his spell at Pittodrie was finding the net twice in the 1988 Scottish League Cup Final at Hampden Park, but opponents Rangers pulled ahead in the final minutes to win the contest 3–2. As he moved into his thirties, he was featuring in the Dons' first team only infrequently, but in late 1989 he was then signed for Rangers, reigning Scottish champions and Dodds' boyhood favourites. The move was surprising given that Rangers manager Graeme Souness had previously likened Dodds to 'a basketball player' when he played against them for Aberdeen. Having been brought to Ibrox primarily as squad cover, Dodds played only a small number of first team matches, sometimes filling in in defence or midfield, before retiring from playing in 1991. Rangers retained his registration, and due to injuries and foreign player restrictions he was named among the substitutes for a UEFA Champions League match at home to Olympique Marseille in November 1992.
International career
Dodds scored on his international debut for Scotland in a 2–0 win over Uruguay in September 1983.
Coaching career
After retiring as a player, Dodds was appointed to Rangers' coaching staff – as first team coach – and continued to be employed there until 1997, being part of the backroom staff under manager Walter Smith, his former teammate at Dundee United, during many of the club's successes in the period. After parting company with the club he decided not to seek further employment in football coaching.