At fourteen years of age Richard joined the Royal Navy, in time rising to the rank of admiral. Like his father and two of his brothers Richard was a celebrated painter, who illustrated various ports and naval scenes.
In 1844, he married Frideswide Maria Moore Smyth (1819–1885) of Portlick Castle, Westmeath. They had two daughters, Annie L. Beechey and Frideswide F. Beechey (1851–1919). In 1882 Frideswide was the first women to win a prize as a composer of chess problems and her book Chess Blossoms was published in 1883 followed by Chess Fruits in 1884, which she co-wrote with her husband Thomas B Rowland 1850-1926. He was also a noted chess player and columnist
In 1888, some three years after the death of his wife, Richard married Frances, daughter of the Rev. Annesley Stewart of Trinity College, Dublin.