This achievement is for getting an A rank on all 5 missions of a single stage.
I found Iron Gate (Eggman's first stage) to be the easiest for this achievement. It's a simple level layout, ring boxes aplenty, and lots of enemies (many of which don't even attack you).
The 1st, 4th, and 5th missions are graded on score, which is not a problem at all as long as you lock onto as many enemies as you can before firing. (You know the lock-on period is almost up when your laser pointer turns blue. If it runs out, everything you locked onto will be unmarked and unharmed.) Also, just keep moving, you don't need to stop anywhere except for on the two elevators. (though if you're careful, you can jump off of them and hover down to the door to save time, but it is not necessary at all.)
The 2nd and 3rd missions are graded on time, based on how quickly you collect 100 rings or find the lost chao, respectively. For these, just ignore any enemies that are not blocking you or shooting at you and just go for the objective.
Here are the criteria you need to meet for each A Rank:
Mission 1: Reach the goal: 20000 points
Mission 2: Collect 100 rings: 1:35
Mission 3: Find the lost chao: 2:00
Mission 4: Reach the goal within 4 minutes: 20000 points
Mission 5: Clear Hard Mode: 19000 points
Remember to keep moving, as your time bonus at the end of the stage can help a lot. Credit to MarioSunshineSprites for the videos