Farrington graduated from Huntsville High School in 1891 and Sam Houston Normal Institute (now State University) in 1893. He later earned a degree from Vanderbilt University in Nashville (BS, 1897) before marrying and returning to Huntsville in 1901. He taught chemistry at SHNI from 1901 to 1944. Farrington was also a life-long member of the Methodist Church, where he taught Sunday School for college students for 35 years, and was a member of the Walker County Chamber of Commerce.
The Farrington Science Building on the SHSU campus (built 1959) was named in his honor.