Born in Saxony, Germany to a family of eight children. His father was the director of the High School and a musical composer and orchestra conductor. Fincke, "remarkably proficient" in mathematics, left school at fifteen and worked as an accountant, and devoted his spare time to scientific studies. In Frankfurt, at age 25, he came down with typhoid fever. While in hospital he met the Reverend William Taube, a follower of Hahnemann, and was introduced to homeopathy. He studied homeopathy with Rev. Taube, and learned how to make his own remedies.
In 1851 he attended a meeting of the Central Society of Homeopathic Physicians in Leipzig, and met Von Boenninghausen, who convinced him to go to America- the "promised land" of homeopathy. He arrived in New York in August, 1852, entered the University Medical College of the City of New York, and received his MD degree in March, 1854. Dr. Valentine Mott, one of his teachers said of him, "I have never known him surpassed among the thousands I have had an opportunity to examine."
Upon graduation he settled in Brooklyn and began a long friendship with Drs. P. P. Wells and Carroll Dunham. His relationship with Dunham became strained after Dunham's 1870 "Freedom of Medical Opinion and Action" address to the AIH, and "...aroused differences of opinion which led to his partial estrangement from Dr. Fincke as well as from other staunch Hahnemannians of the day, who foresaw it as a tendency to break down the barriers between the schools at the expense of homeopathic integrity."
In 1865 Fincke wrote a text, On High Potencies. Three years later he married Maria Catherine Ficht. In 1896 he was elected president if the International Hahnemannian Association. In 1897 Fincke, with the Brooklyn Hahnemannian Union, did the first proving of x-ray.
He began producing high-potency remedies in the late 1860's, and many of his remedies are still in use to this day.
According to Stuart Close, Fincke was "reclusive" and "little known to the members of the profession in his own city... He was of a very sensitive, retiring disposition, and slow to give his confidence, but once bestowed, his friendship never wavered."
A naturalized American citizen, he kept portraits of Washington and Lincoln over his mantelpiece. In politics he was a staunch Republican. In religion he was a lifelong member of the German Evangelical Lutheran Church. Close relates that Fincke flew the American flag over his summer home at Bella Sylvia in Pennsylvania, raising it every morning and taking it in every evening.
He was an accomplished musician on the piano and the horn.
His death, relates Close, was peaceful. Several days before the end he began to have visions of all the bottles in his office becoming two feet high. "Most peculiar." he said. "Bottles with heads, all in a row, bending forward, and all going in one direction- toward the west- toward the setting sun."