He was educated at a private school, and for some years assisted his father in his business, but in 1824 gave up trade and went to the continent to study philosophy and literature.
After spending four years principally in Italy and Germany, he returned for a winter's term at the University of Edinburgh, went again to the continent, and eventually settled in 1830 at Munich to study philosophy under Friedrich Schelling and Old German under Johann Andreas Schmeller and Hans Ferdinand Massmann.
In 1839 he collated the Codex Argenteus at Uppsala, and in January 1840 he formed the plan of his Icelandic-English Dictionary, starting work by April.
After some false starts and the temporary loss of some of Cleasby's papers, Guðbrandur Vigfússon in 1864 took over the Dictionary, and George Webbe Dasent lobbied the Clarendon Press for backing.
The work was eventually completed in 1873, and published with a preface by Henry Liddell, and an introduction and memoir of Cleasby by Dasent.