After playing for several amateur teams in the Birmingham area, Jenkyns joined Small Heath (later renamed Birmingham) in 1888. In 1889 the club first joined the Football Alliance and in 1892 became a founding member of the second division of the Football League. He had now made his debut for Wales and was club captain. In 1894 he made promotion to the first division as captain of Small Heath and defeated Darwen 3-1 in a friendly.
Known as one of the toughest defenders of his era, he was sent off four times at a game for Small Heath, at a time when such occurrences were extremely rare. His career in the Kleine Heide ended in March 1895 when
he was sacked after an incident in the derby when he tried to attack two spectators on orders from the field. Jenkyns moved to London in April 1895 and joined the Woolwich Arsenal, which had joined the Second Division less than two years ago. Immediately named Arsenal captain, he made a name for himself in Arsenal history by becoming the club's first ever international after winning a game against Wales against Scotland on March 21, 1896.
Jenkyns played downtown (then a midfield position), scored six goals in 27 games for Arsenal and was one of the club's star players. However, his stay with the Gunners did not last long; in the summer of 1896 he moved to Newton Heath (later Manchester United). On September 1, 1896, Jenkyns made his debut for the club against Gainsborough Trinity. He spent two seasons with the Manchester Gunners and took over the position of captain from James McNaught and helped them to second place in the second division in 1897. In November 1897 he left the club and went to Walsall, where he spent five years before moving to Coventry City.
After his retirement, Jenkyns ran a public house in Moxley, The George Inn, before going to the police.
​