He was educated at Westminster and Trinity College, Cambridge. He was awarded his BA degree in 1628, then MA in 1632, and became a major fellow of his college in the same year.

He soon gave promise as a writer of comedy. Ben Jonson, not an easily satisfied critic, adopted him as one of his "sons." He addressed three poems to Jonson, one on the occasion of his formal "adoption," another on the failure of The New Inn, and the third an eclogue, describing his own studies at Cambridge. He lived with his father at Little Houghton, Northamptonshire for some time, and afterwards with William Stafford of Blatherwycke Hall, where he died aged 30. He was buried in Blatherwycke church on 17 March 1635 and his epitaph was written by Peter Hausted, the author of The Rival Friends, on his monument, commissioned by Sir Christopher Hatton.

Randolph's reputation as a wit is attested by the verses addressed to him by his contemporaries and by the stories attached to his name. His earliest printed work is Aristippus, Or, The Joviall Philosopher. Presented in a private shew, To which is added, The Conceited Pedlar (1630). It is a gay interlude burlesquing a lecture in philosophy, the whole piece being an argument to support the claims of sack against small beer. The Conceited Pedlar is an amusing monologue delivered by the pedlar, who defines himself as an "individuum vagum, or the primum mobile of tradesmen, a walking-burse or movable exchange, a Socratical citizen of the vast universe, or a peripatetical journeyman, that, like another Atlas, carries his heavenly shop on shoulders." He then proceeds to display his wares with a running satirical comment.

The drama, The Jealous Lovers, was presented by the students of Trinity College, Cambridge, before the king and queen in 1632. The Muse's Looking-Glass is hardly a drama. Roscius presents the extremes of virtue and vice in pairs, and last of all the "golden mediocrity" who announces herself as the mother of all the virtues. Amyntas, or The Impossible Dowry, a pastoral printed in 1638, with a number of miscellaneous Latin and English poems, completes the list of Randolph's authenticated work. Hey for Honesty, down with Knavery, a comedy, is doubtfully assigned to him. Randolph has been proposed as the author of the anonymous manuscript play, The Fairy Knight, though the attribution has not won much approval from critics.

His works were edited by WC Hazlitt in 1875.