2. To accomplish something successfully; perform at a standard or above a standard level.
3. Etymology: from Late Latin ad caput (venire, "to come"); both the Old French and Late Latin phrases meaning literally, "to come to a head", from Latin caput, "head".
To achieve something is to bring plans and actions to a head. This is the literal meaning of the ancestor of our word achieve, which was borrowed from the French in Norman times. They made the verb achever out of the preposition a-, "to", and chief, "head": "to bring to a head".
Old French chief comes, in turn, through a thousand years of gradual changes, which only the strongest (accented) syllables survived, from Latin caput, "head". The original sense, "bring to a head", easily acquired the present significance, "bring to a successful conclusion, make a success" of the task at hand.