Beeching was born in Hastings, Sussex, England, on May 8, 1922, and died in Palma de Mallorca, Spain, on December 27, 2001. He served in the Fleet Air Arm during World War II. He married Gwendoline (Inez) Mathews in the early 1940s. This marriage produced one son, Matthew (1946). They were divorced in 1948. He married author and translator Amy Brown in 1950 and had a son, John, and a daughter, Laura. They were divorced in 1970, and he shared the rest of his life with Charlotte Mensforth, the painter. He had four children.

In 1956, he suffered serious chest injuries in an automobile accident. As a result, he had severe respiratory problems that forced him to live in warmer climates.

Beeching's poetry is considered moving, original, clear-sighted, compressed, and funny. This was a view expressed by the editor of Qualm in 2003, a high opinion shared by the editors at Penguin circa 1970, and reflected in his obituary in The Independent thirty years later, whose author speaks also of Beeching's 'disciplined metre, subtle half-rhyme and a luxuriant syntax which expressed at times distinctly "difficult" metaphysical concerns'. His writing in old age was perhaps at least as strong and trenchant as that of any of his peers of a similar age.

Although he continued to write until his death, during the second half of his life his work fell into neglect. This neglect was partly attributable to his having to live, because of his damaged lung, abroad in drier climates, including Greece, Turkey, Guatemala, Lucca, Genoa, Menton, and Majorca. It was a life of near-poverty in tiny apartments.

He was published in Penguin Modern Poets No. 16 in 1970, and near the end of his life brought out a collection, Poems 1940-2000 (Art Ojo Nuevo). He was a novelist and writer of historical books, but stated "Poetry is my avocation; the other forms of writing are a means of livelihood".

Beeching translated poetry from French and Spanish, and wrote several plays for the London stage. He contributed to The New York Times and The Times.

The Arts Council of Great Britain gave him their Award to a Living Artist in 1967, and he was later granted a Civil List pension for "services to literature".