In early stages of English, only c was used, even in native Germanic-origin words: consider Old English cynd (modern kind), cyning (king), cepan (keep). The letter k was introduced to English writing in the Middle English period, i.e., after the 11th century.
Old High German, on the other hand, seems to have used both k and c interchangeably, before standardizing on k in the modern period.
(Disclaimer: I am a linguist, but this isn't my area of expertise; I don't know much detail about the history of writing systems in the Germanic languages and I could be mistaken here.)