Leslie was born on March 29, 1821, in Ipswich, England as Henry Carter, the son of Joseph Carter, the proprietor of a long-standing and prosperous glove manufacturing firm. He was educated in Ipswich and he then trained for commerce in London. As a boy on his way to and from school, he passed a silversmith's shop whose workers he took a detailed interest in, especially those who engraved designs and letters upon various articles of silver and gold. He took note of the tools that were used and the manner of using them and acquired the necessary tools to do the work himself. At the age of 13, he did his first wood engraving of the coat of arms of his home town.