Archer was brought up during the First World War in Walthamstow, East London. A painter and etcher, Frank attended the Brighton School of Art and later the RCA. Archer was commended for his talents and awarded the Rome Scholarship in 1938. Archer left England for his studies in Italy. However this was short lived as he had to return not long after due to the outbreak of war. Archer began teaching at the Kingston College of Art under the distinguished painter-etcher Wilfred Fairclough. Archer went on to replace Fairclough.

He had a passion for music and musicians having been a violinist himself, and  featured musicians in his work. One can see the light and colour dancing in his pieces. Archer became a Fellow of the Royal Watercolour Society after having joined as an Associate in 1972.