Meek was the son of Job Meek and Marianne McKain. The Dictionary of Australian Artists (ed. Joan Kerr, 1992) gives his year of birth as 1815. Meek may initially have emigrated to New South Wales, but by 1847 he was a fisherman in the Melbourne area. In 1851 he went to Ballarat as a gold miner and also ran the first general store, a rather unsuccessful enterprise. The first of his drawings still extant was Meek's Store, Ballarat 1851 and he went on to produce other drawings of life on the goldfields in 1851-1852. In 1861 Meek won a second-class certificate at the Victorian Exhibition for his pen-and-ink Map of Australia. He also sketched Melbourne's major buildings and produced a Tablet of Victoria containing information on the progress of the colony. In 1869 Meek's The Resources of the Western District was published.
Meek's Chronological Tree of the History of Victoria 1836-1872 was photolithographed by John Noone in the Crown Lands Office and published by John Patten. His Poster Commemorating the Separation of Victoria from New South Wales was printed by Hamel and Co. Meek visited Sydney in February 1863 and exhibited a lithograph of his Historical and Descriptive Atlas of the British Colonies in Continental and Insular Australia (1861), a general map of Australia and a Shakespearian Memorial. The Atlas won a gold medal at the 1862 London International Exhibition. Meek was considered 'the best penman in Australia' after producing the Ballarat Historical Gumtree (see Dictionary of Australian Artists). The Dictionary also asserts that Meek 'later settled in New Zealand, where he died'.
Meek himself gives a different version, stating in his Preface to The Creation (1898) that shortly after the first publication of his epic poem The Creation in 1874 'myself and family left Victoria for New Zealand on board the "S.S.Hero", Captain Logan, taking with me the books I had published, and settled in Auckland for a time.' Meek indicated that he returned to Ballarat in 1891.While in New Zealand he continued to publish poetry and on his return wrote Zealandia's Annie Macquaid : only a stewardess : Albion's Grace Darling (189?). The Victorian Registry of Births Deaths and Marriages indicates that Meek died in Victoria.