Felch was born in Limerick, Maine. He was left an orphan at the age of three and lived with his grandfather Abijah Felch, a veteran of the American Revolutionary War. He attended the Phillips Exeter Academy, Exeter, New Hampshire and graduated from Bowdoin College, Brunswick, Maine in 1827. He studied law and was admitted to the bar and practiced in Houlton, Maine from 1830 to 1833.

He moved to Monroe, Michigan in 1833 and continued the practice of law. He was elected three times to the Michigan State House of Representatives, serving from 1835 to 1837. He was appointed State bank commissioner in 1838 and resigned in 1839. He was state auditor general for a short time in 1842 before being appointed associate justice of the Michigan Supreme Court in 1842, where he served until his resignation in 1845, after being elected Governor. He served as Governor of Michigan from 1846 to 1847 and resigned on March 3, 1847 after being elected as a Democrat to the United States Senate. He served in the 30th, 31st and 32nd Congresses, from March 4, 1847, to March 3, 1853.

In March 1853, he was appointed by U.S. President Franklin Pierce to be part of a commission to settle Spanish and Mexican land claims in California arising from the Treaty of Guadalupe Hidalgo which ended the Mexican-American War and served as president of the commission until 1856. He returned to live in Ann Arbor, Michigan where he resumed his law career and served as the Tappan Professor of Law at the University of Michigan from 1879 to 1883.

He died in Ann Arbor and is interred in Forest Hill Cemetery.

Felch Township, Michigan, a station on the Chicago and North Western Railway in Dickinson County, is named in his honor, as is Felch Street in Ann Arbor, Michigan.