To get this achievement you need to complete every wave (1 through 20) of any Last Stand stage. If you fail out, then you can retry starting on the wave you failed on. Difficulty does not matter so Rookie would be the best way to go.
It is recommended to try to get this achievement while also going for the Last Man Standing achievement because completing 20 waves is going to take up to an hour anyway so you may as well kill two birds with one stone.
To dispel any confusion, starting at Wave 15 will not count towards this achievement. Starting from Wave 15 and finishing Wave 20 will, however, count towards Last Stand Completionist and Last Stand Master.
Update: This achievement seems to be somewhat glitchy, but so far all glitches have worked positively. Some people have unlocked the achievement before Wave 20, possibly because they had started on a later wave, fought through some waves, failed, and restarted from Wave 1. Either way, it would be best to start from Wave 1. (Thanks to Agonys and CaptainCombOver)
If you have any suggestions on how to make this solution better, please let me know in the comments.