Little is known of his early life. The son of Henry Jenkins, a carpenter who occasionally made musical instruments, he may have been the "Jack Jenkins" employed in the household of Anne, Countess of Warwick in 1603. The first positive historical record of Jenkins is amongst the musicians who performed the Masque The Triumph of Peace in 1634 at the court of King Charles I.
The English Civil War that broke out in 1642 forced Jenkins, as it did many others, to migrate to the rural countryside. During the dark days of the 1640s he was employed as music-master to two Royalist families, the Derhams at West Derham and Harmon L'Estrange of Hunstanton. He was a friend of the composer William Lawes (1602-1645), who was shot and died in battle at the siege of Chester.
Around 1640 Jenkins revived the In Nomine, an archaic form for consort of viols, based upon a traditional plainsong theme. He wrote a notable piece of programme music consisting of a pavane and galliard depicting the clash of opposing sides, the mourning for the dead and the celebration of victory after the siege of Newark (1646).
In the 1650s Jenkins became resident music-master of Lord Dudley North in Cambridgeshire, whose son Roger wrote his biography. It was in these years, during the Commonwealth under Oliver Cromwell, in the absence of much competition or organised music-making, that Jenkins took the occasion to write more than 70 suites for amateur household players.
Jenkins played the lute and was a virtuoso upon the lyra viol. After the Restoration he obtained a place as a musician to the Royal Court. The aged Jenkins played the lyra viol for King Charles II, who wryly complimented him that he did "wonders on an inconsiderable instrument". Roger North wrote:
Jenkins retired under the patronage of Sir Philip Wodehouse of Kimberley, where he met Sir Thomas Browne. Although the musicologist Wilfred Mellers claimed that J.S. Bach's Orchestral Suites No. 3 and No. 4 in D major (BWV 1068-69) recalled the sensibility of Sir Thomas Browne, Jenkins's music is much closer historically to an aural representation of the sensibility of this physician-philosopher.
Something of Jenkins's own temperament is indicated by his setting the religious poetry of George Herbert to music. Like Haydn, he was a pious, reticent, and private person. Workmanlike and industrious in composition, he wrote dances "by the cart-load", according to North.
Jenkins was a long-active and prolific composer whose many years of life, spanning the time from William Byrd to Henry Purcell, witnessed great changes in English music. He is noted for developing the consort fantasia for viols, being influenced in the 1630s by an earlier generation of English composers including Alfonso Ferrabosco the younger, Thomas Lupo, John Coprario and Orlando Gibbons. Jenkins composed numerous 4, 5, and 6 part Fantasias for viol consort, Almans, Courants and Pavanes, and he breathed new life into the antiquated form of the In Nomine. He was less experimental than his friend William Lawes; indeed, Jenkins's music was more conservative than that of many of his contemporaries. It is characterised by a sensuous lyricism, highly skilled craftsmanship, and an original usage of tonality and counterpoint.
His biographer North wrote of him:
Jenkins is buried in the nave of St. Peter's church, Kimberley, Norfolk, with this inscription: