SONIC THE HEDGEHOG

Of the three characters, I find this to be the easiest all-S achievement. That's not to say, however, that you'll be able to get this with ease.

For the main acts, remember to use Sonic's custom actions (gems) to help you out. The yellow gem will give you a shield that attracts rings and the red gem will slow everything around you by 50%. In most stages, you'll want to finish somewhat quickly, defeat most enemies, and hold onto a few hundred rings (when possible).
One of the most difficult levels to S-Rank is Crisis City. There's a high chance you won't make it through the Mach Speed section at the end of the level without getting hit. On both Normal and Hard, I recommend focusing on reaching this section as fast as possible. In this area, focus on getting as many rings as you can without getting hit. If you manage to get at least one 1-up after each restart (if you get it), you won't have to worry about a Game Over (though it will probably frustrate you when constantly getting hit).
For each boss, the red gem helps a lot with dodging and attacking.. Use this with the yellow gem to get every ring, thus boosting your score.

Addendum: If you're having trouble with White Acropolis, here's a trick that also works with Shadow and Silver. If you break a searchlight (not the structure itself), you can go a certain distance away from it and return. The light will re-appear and you can continue to destroy it for 1000 points each time.