First, you must go into the games mode and select Sonic the Hedgehog. The aim of this game is to get rings and avoid the spikes and bombs as you collect rings. After some time (or rings?), a big ring appears which takes you to a bonus stage. Within the bonus stage, you must collect all the rings. When you do, you get a chaos emerald. When you collect 7 emeralds, the achievement is yours.
To start, I chose Shadow the Hedgehog. Next, I followed up with sweeping left to right motions across the court, not necessarily going for every single ring. The sweeping motion keeps the spikes spread out, and paces the time it takes for you to get rings. I noticed that more 1ups appeared when I took my time and made it look like I needed help. Be sure to stay on the outsides and bottom as well in case bombs appear. What I mean by that is, the game starts with forming groups of 3 rings, then groups of 4 as you progress, then groups of 5, and finally by the 8th (and every other divisble-by-8 level) level groups of 5 rings that move around the court. Well sometimes, instead of the appropriate grouping of rings, one of the rings is replaced by a stationary bomb that outlasts the grouping of rings, so it needs to be avoided in order to get all the rings.
JUST PLAY CAUTIOUSLY, even if you have many lives. Don't try going for every single ring if it's too dangerous, especially when the robots start launching their spikes without hesitation. Just swing around the court in sweeping left to right movements to dodge, and collect what you can as you do so.
Doing this, I got the final emerald around level 14 or 15. Trying this strategy I got the achievement on my first try, versus just going for rings which I lost about 6 times before switching strategies.
Good luck, and extra good luck going for the AAA achievement X(