Bloxam's father was an assistant master at Rugby School. He was educated at Elborow School before attending Rubgy himself between 1813 and 1820. In 1821 he was articled to a solicitor but he did not find success in the profession when he went into practice on his own account. In 1831 he became clerk of the court and held the post for 40 years.
He is remembered as an antiquarian on Rugby and the local area. His work was published in two books (including Principles of Gothic Architecture) and many journal articles though many of his conclusions are now thought doubtful his collection of archaeological finds still exists.
He lived in what is now the Percival Guildhouse, while his brother ran a boarding school next door in what became the public library. A new library replaced the old one in 2000 and a life-size statue of Bloxham engaged in his archaeological work greets visitors to the Rugby museum located in the new Library complex.
Bloxam is the sole source of the story of William Webb Ellis picked up the ball during a game of football at Rugby School thus originating the game of Rugby football. In October of 1876, he wrote to The Meteor, the Rugby School magazine, that he had learnt from an unnamed source that the change from a kicking game to a handling game had "..originated with a town boy or foundationer of the name of Ellis, William Webb Ellis".
In December of 1880, in another letter to the Meteor, Bloxam elaborates on the story:
"A boy of the name Ellis - William Webb Ellis - a town boy and a foundationer, .... whilst playing Bigside at football in that half-year, caught the ball in his arms. This being so, according to the then rules, he ought to have retired back as far as he pleased, without parting with the ball, for the combatants on the opposite side could only advance to the spot where he had caught the ball, and were unable to rush forward till he had either punted it or had placed it for some one else to kick, for it was by means of these placed kicks that most of the goals were in those days kicked, but the moment the ball touched the ground the opposite side might rush on. Ellis, for the first time, disregarded this rule, and on catching the ball, instead of retiring backwards, rushed forwards with the ball in his hands towards the opposite goal, with what result as to the game I know not, neither do I know how this infringement of a well-known rule was followed up, or when it became, as it is now, a standing rule."