Ache is one of the great embarrassments of English spelling. The word is native, from Old English acan, "to suffer continued pain," and for most of its life it was spelled and said with a plain k sound. English kept a neat pair, much like speak and speech: the verb was ake, the noun was ache. Then the noun's pronunciation drifted to match the verb's, and around 1700 the spelling of both was changed to ache — on the mistaken belief that the word came from Greek akhos, "pain, distress."
It didn't. Ache is Germanic to the core, and the Greek word is at best a distant cousin, related to awe rather than an ancestor. So the ch here is a learned costume put on a homely old word by editors who wanted it to look classical, and English has been stuck with the mismatch ever since. It is the clearest possible case of a spelling that records what somebody believed about a word rather than where the word actually came from.
ache (v.) Middle English aken , from Old English acan "suffer continued pain," from Proto-Germanic *akanan , which is perhaps from a PIE root *ag-es- "fault, guilt," with apparent cognates in Sanskrit and Greek, which itself is perhaps imitative of groaning. Originally the verb was pronounced "ake," the noun "ache" (as in speak / speech ). The noun changed its pronunciation to conform to the verb, but the spelling of both was changed to ache c. 1700 on a false assumption of a Greek origin (specifically Greek akhos "pain, distress," which rather is a distant relation of awe (n.)). Related: Ached ; aching . ache (n.) "continuing pain," early 15c., æche , ece "an ache, pain," from Old English æce , from Proto-Germanic *akiz , from same source as ache (v.), which see for the unusual evolution of spelling and pronunciation. also from early 15c.