Raper studied chemistry at Yorkshire College Leeds, and graduated with first class honours in 1903. He then worked at the Lister Institute in London before returning to Leeds to read for a medical degree. He qualified in 1910, and was awarded the D.Sc in the same year. He was lecturer in pathology at Toronto University from 1910-13, but in 1913 returned to Leeds as lecturer in physiology and biochemistry. Raper was appointed professor of physiology at Leeds in 1918, having worked during the war at the Royal Engineers Anti-Gas Establishment. In 1923 he was elected to the Brackenbury chair of physiology at the University of Manchester. He remained in this post until 1946, when he transferred to the chair of chemical physiology. He was also honorary advisor in physiology at MRI from 1923-51 and served as dean of the University Medical School on several occasions. Raper was member of the Medical Research Council from 1933-9, and of the General Medical Council from 1943-51. He was president of the Manchester Medical Society in 1938. Raper was a member of MRHB from 1948 until his death, and was chairman of the Manchester Dental Hospital in the same period. He was elected FRS in 1929 and was a member of is Council from 1939-41.
Raper's research interest was in biochemistry, particularly the metabolism of fats, the formation of melanin and tyrosinase-tyrosine reactions. He played a vital role in the development of a clinical research laboratory at MRI. Raper was considered one of the leading biochemists of his day.