early 14c., "to perform, execute, accomplish;" late 14c., "gain as a result of effort," from Old French achever (12c.) "to finish, accomplish, complete," from phrase à chef (venir) "at an end, finished," or Vulgar Latin *accapare, from Late Latin ad caput (venire); both the French and Late Latin phrases meaning literally "to come to a head," from ad "to" (see ad-) + stem of Latin caput "head" (from PIE root *kaput- "head").
The Lat. caput, towards the end of the Empire, and in Merov[ingian] times, took the sense of an end, whence the phrase ad caput venire, in the sense of to come to an end .... Venire ad caput naturally produced the Fr. phrase venir à chef = venir à bout. ... From this chief, O.Fr. form of chef (q.v.) in sense of term, end, comes the Fr. compd. achever = venir à chef, to end, finish. [Auguste Brachet, "An Etymological Dictionary of the French Language," transl. G.W. Kitchin, Oxford, 1878]
Related: Achieved; achieving.