As a child, Kelly suffered from Legg–Calvé–Perthes disease, but recovered to win a move from non-league Alvechurch and turn professional at Walsall in 1983. He was named in the 1986–87 Third Division PFA Team of the Year, and scored a hat-trick in the 1988 play-off final to secure the club promotion out of the Third Division. He was sold to West Ham United for £600,000 in August 1988, but struggled for form before being sold on to Leicester City for half that sum in March 1990. He then moved on to Newcastle United for a fee of £250,000 in December 1991. He helped Newcastle to avoid relegation out of the Second Division in 1991–92, before being named as the club's Player of the Year as United won promotion into the Premier League as champions of the newly-renamed First Division in 1992–93.
He remained in the First Division however, having been signed by Wolverhampton Wanderers for a £750,000 fee in June 1993. He top-scored for Wolves in 1994–95, before he was purchased by Sunderland for £900,000 shortly before they won promotion as champions of the First Division at the end of the 1995–96 season. He dropped back into the First Division after joining Tranmere Rovers for £350,000 in 1997, and spent three seasons with Rovers, playing on the losing side of the 2000 League Cup final. He went on to finish his career following brief spells with Sheffield United, Motherwell, Mansfield Town, and Derry City, and helped Derry to win the 2002 FAI Cup in his final match as a player.
Kelly was diagnosed with Legg–Calvé–Perthes disease at the age of five, and at one stage his left leg was four inches shorter than the right leg, and he was on crutches until the age of ten. He played football for Bartley Green Boys regardless, and was attached to West Bromwich Albion before he was released by Albion, at which stage he worked as a trolley porter at Cadbury's whilst representing the youth team at non-league side Alvechurch. He joined Walsall following a successful trial in 1981, and turned professional at the club under Alan Buckley two years later. He scored three goals in six Third Division games in the 1983–84 season, before winning a regular first team place in the 1984–85 campaign, scoring 13 goals from 41 appearances. He provided ten goals from 28 league games in the 1985–86 campaign, before his career took off under new manager Tommy Coakley; he top-scored with 26 goals from 55 appearances as the "Saddlers" posted an eighth-place finish in 1986–87, and was named in the Third Division PFA Team of the Year. He then scored 30 goals from 54 matches in the 1987–88 season as Walsall won promotion via the play-offs in 1988; he scored a hat-trick in the replay of the 1988 play-off final as Walsall beat Bristol City to achieve promotion. He went on to have a week-long trial at Bayern Munich, but did not join the club despite Uli Hoeneß reportedly comparing him to Denis Law. He scored a total of 82 goals in 190 league and cup appearances over the course of five seasons at Fellows Park, and the transfer fee received for him remains a club record.
His exploits attracted the attention of West Ham United, who signed him for a fee of £600,000 in August 1988. His stay at the Boleyn Ground proved to be largely unsuccessful however, as he scored only six goals in 25 First Division games as John Lyall's "Hammers" suffered relegation in 1988–89. Kelly also failed to find form for the club under new boss Lou Macari in the Second Division, scoring just two goals in 24 matches of the 1989–90 campaign.