This is less of a guide, and more of a clarification. For anyone wondering, while the achievement is stackable (meaning you can complete all of the chapters on Normal, but still unlock this achievement, also), it does NOT let you mix-and-match. Allow me to explain:
I completed the first three levels (1-1, 1-2, and 2-1) on Normal. Then, for some reason, I decided to play on Amateur. I completed the rest of the game on Amateur and got achievements for each of those levels, but I didn't get any achievement with the description of "Complete all chapters on..." So I thought that the Normal completions on the first three levels must have NOT counted toward the "all chapters" on Amateur.
By going back and completing 1-1, 1-2, and 2-1 on Amateur, the achievement finally popped.
To sum up, you must complete all of the chapters on the same difficulty in order to unlock these "all chapters" achievements. You can check your records to see which levels are missing, if the achievement didn't pop when you thought it would.