He was born in New Haven, Connecticut to James Osgood and Elizabeth Badger. He studied painting in Boston, Massachusetts. After his marriage to poet Frances Sargent Locke he continued his art education at the Royal Academy in London. Upon returning to America he settled in New York City, where he was made an associate of the National Academy of Design. In 1849 he went to California where he stayed nearly a year, prospecting for gold and painting portraits in San Francisco.
His wife died of tuberculosis in 1850. Osgood designed her memorial at Mount Auburn Cemetery in Cambridge, Massachusetts, where she was buried. Inspired by her poem "The Hand That Swept the Sounding Lyre", Osgood designed a 15-foot memorial: a white marble base topped by a bronze lyre crowned by a laurel wreath. Four of the five strings of the lyre were designed as cut (symbolizing his wife and three daughters, who were all dead by 1851). On his death, he asked his second wife to cut the fifth string.