Schrempf was born in Besigheim and was raised by his mother's brother. He attended schools in Ingersheim, Boll and Horrheim. After the aspirant exam in March 1871, he was a student at the private seminar Tempelhof in Oberamt Crailsheim from 1872 to 1875, and then was a seminary teacher there until 1879. After that he was a teacher at the boys' school of Professor Pfleiderer and in the community grammar school in Korntal until 1890. Subsequently, he was a contributing editor of the conservative daily Deutsche Reichspost in Stuttgart, and from spring 1892 to 1909 lead editor. From 1890 he was also secretary of the conservative Württemberg Party. On 1 December 1912, Schrempf retired. He died in Stuttgart.