Jenkins was born and grew up in Jersey Marine, Wales. He was a youth goalkeeper for Afan Nedd Schools, Llandarcy and Afan Lido at the Welsh League level. He played in the youth systems of Bristol Rovers and Swansea and earned one Boys Club of Wales cap playing against Scotland, at 14. He's a lifelong Swansea fan, recalling being taken by his father, Gordon Jenkins, to watch Swansea play Cardiff at Ninian Park as a boy. "We’d park away from the ground, and sneak in to see the game," he said in an interview, because "it was quite hostile if you were supporting Swansea."
Swansea City
Jenkins was chairman, shareholder and chief executive of Swansea City A.F.C.. Being part of a consortium that "ousted" then-owner Tony Petty and with the backing of the Swansea City's Supporters Trust, he was appointed chairman in January 2002, a position that he took, as he states, because, "I was the dullest one." During his chairmanship, the club's finances turned to profitable and the club progressed on through to the Premier League. In 2013, Swansea City won the League Cup. It was the first time the League Cup had been won by a non-English club. During the 2013–14 season, Jenkins sacked manager Michael Laudrup in February 2014 and replaced him with former player Garry Monk, stating, "we had to remove the constant uncertainty surrounding the club and Michael's long-term future with us." Laudrup, in his first season with the club, had led Swansea to 9th place in the Premier League and to "League Cup glory, their first ever major trophy," while taking them to next season's Europa League campaign, a campaign crowned with a 3–0 away win over Spanish top-flight side Valenciain September 2013. After his sacking, the club and Laudrup reached an out-of-court settlement whose details were not disclosed. In April 2016, a "controlling" portion of the club's shares were sold to American investors. At the end of the 2017–18 season, the club was relegated to the second tier. On 2 February 2019, Jenkins resigned as chairman amid increasing criticism over the club's sale to the American consortium in 2016 and the club's subsequent relegation from the Premier League. He stated, "...the current atmosphere within the football club on and off the football field saddens me very much and I find it very difficult to fight on in a football club I love but can no longer control."
Towards the end of May 2020, it was reported that Jenkins was an interested party in buying Charlton Athletic F.C. after the club spent a turbulent few months under the ownership of East Street Investments.