From Middle English aken, and ache, from Old English acan (from Proto-Germanic *akaną) and æċe (from Proto-Germanic *akiz), both from Proto-Indo-European *h₂eg-. Cognate with Low German aken, äken, North Frisian akelig, æklig, West Frisian aaklik, Dutch akelig.
The verb was originally strong, conjugating for tense like take (e.g. I ake, I oke, I have aken), but gradually became weak during Middle English; the noun was originally pronounced as as spelled (compare breach, from break). Historically the verb was spelled ake, and the noun ache (even after the form started to become common for the noun; compare again break which is now also a noun). The verb came to be spelled like the noun when lexicographer Samuel Johnson mistakenly assumed that it derived from Ancient Greek ἄχος due to the similarity in form and meaning of the two words.
From Middle English ache, from Old French ache, from Latin apium. Reinforced by modern French ache.