Beck was born in England in 1749. He was employed as an instructor in mathematics at Woolwich from 1776, but was afterward dismissed. He emigrated to the United States in 1795, and was employed in painting pictures. One of his paintings, The Great Falls of the Potomac (1796) was purchased by President George Washington. Washington hung the painting in his parlor at Mount Vernon, where it remains today.

Beck also wrote short poems, made poetic translations from Anacreon, Homer, Virgil, and Horace, and in 1812 published Observations on the Comet.

In 1795, he served as a scout in General Anthony Wayne's campaign against the Native Americans.

With his wife, Bethany Marcus who was also an artist, Beck conducted a female seminary in Lexington, Kentucky for many years. He died in Lexington on December 24, 1812.