Kelly was born in Lusmagh, County Offaly in 1955. He was educated locally and later attended the Presentation Brothers Secondary School in Birr. It was here that he first made a name for himself at the game of hurling. Kelly later became a member of the Garda Síochána.
Kelly played his club hurling with his local Lusmagh club. He enjoyed some success culminating in the winning of a senior county title. This was the first, and since then the only, county title that Lusmagh had ever won.
Kelly first came to prominence on the inter-county scene at a time when it was unfashionable to play hurling for Offaly. He made his debut for the Offaly minor team in 1971 and was part of the county's under-21 set-up a couple of years later. Kelly lined out for the county senior team for the first time in the autumn of 1974 and marked his arrival with a point against Antrim in a National Hurling League game.
Success was slow in coming for Kelly, particularly since Offaly had no great hurling tradition. All this changed in 1980 when Offaly came from nowhere to defeat Kilkenny and capture their first Leinster title. Kelly played a huge role in that amazing victory, however, Offaly's fairytale season ended with a defeat by Galway in the All-Ireland semi-final. In spite of not claiming the All-Ireland title Kelly was still presented with his first All-Star award. Offaly bounced back in 1981 with Kelly capturing a second consecutive Leinster medal. This victory allowed Offaly a straight passage into the All-Ireland final where Galway provided the opposition. Things looked bad for Offaly as the second-half saw Galway take a six-point lead. The men from the West failed to score for the final 23 minutes of the game, allowing Offaly to capture a 2-12 to 0-15 win. Kelly had finally captured his first All-Ireland medal.
Offaly surrendered their provincial crown to Kilkenny in 1982 and 1983, however, the team bounced back in 1984 to win the Leinster title for the third time. A facile win over Galway in the All-Ireland semi-final resulted in Offaly being installed as hot favourites to claim their second All-Ireland. Cork provided the opposition in the championship decider in the first ever meeting of these two teams. Offaly's favourites tag was unjustified as Cork swept them off the field on a score line of 3-16 to 1-12. In spite of this loss Kelly was presented with a second All-Star award. In 1985 Offaly retained their provincial crown, giving Kelly a fourth Leinster title. This victory allowed Offaly to advance directly to the All-Ireland final where Galway provided the opposition. In an exciting game Offaly emerged victorious giving Kelly his second All-Ireland title. The next few years saw Kilkenny reclaim the Leinster title. Offaly returned to their winning ways in 1988 with Kelly capturing a fifth Leinster medal. The subsequent All-Ireland semi-final resulted in a loss for Kelly's side. The following year he added a sixth Leinster medal to his collection. The All-Ireland semi-final produced one of the biggest hurling shocks of all-time. Antrim came from nowhere to defeat the men from the Faithful County on a score line of 4-15 to 1-15. The significance of Antrim's victory was not lost on the Offaly team, who provided a guard of honour for the Ulstermen as they left the field.
In 1990 Kelly won a sixth Leinster medal, however, defeat soon followed in the All-Ireland semi-final. 1991 began well as Offaly captured their first National League title, however, Offaly surrendered their provincial crown to Kilkenny. Kelly played his final championship game for Offaly against Kilkenny in the Leinster semi-final of 1993.
Four years before his retirement from inter-county hurling Kelly had already ventured into hurling management. He broke his leg in 1990 which ruled him out of playing for some time. To stay involved in the game he took charge of the Westmeath senior hurling team. In his first year in charge Westmeath won promotion to Division 2 of the National League. In Kelly's second year in charge his team won the All-Ireland 'B' title in Croke Park.
After his two-year stint with the Westmeath Kelly joined Éamonn Cregan's Offaly backroom team in 1995. He remained a selector for just over a year, having tendered his resignation after the Leinster final defeat to Wexford in July 1996.
At club level Kelly had a two-year spell as manager of Loughmore-Castleiney in Tipperary. In 1997 he devoted his time to the management of Kilcormac-Killoughey. In 1999, he managed the Wexford senior hurling team for one season, ending in a heavy championship defeat to Offaly.
In 2008 Joachim was announced as manager of the Offaly Junior Camogie Team which is their main team. In his first year they reached the All-Ireland Junior Camogie Final. They lost to Clare after a late late goal by Shonagh Enright. However just one year later in 2009 they did in fact capture the All-Ireland Camogie title.
He has been announced as the trainer for Roscommon senior hurling team for 2014, working with manager Justin Campbell, a former Galway hurler.