If you must use the noun "achievement" (rather than the verb "achieve") and find yourself in need of a verb to go with it, there are possibilities but all of them sound a little bit awkward.

Perform, win, score or gain are the sorts of things I have in mind.

e.g. He performed the remarkable achievement of three birdies in a row.

But it would be less wordy to say He achieved three birdies in a row.

As someone has noted achieve and achievement are already strong words.

Of course achievement works well with the verb to be, as in To have climbed Everest is a stunning achievement.