Thanks to
https://www.trueachievements.com/PistolMcAwesome.htm (formerly
https://www.trueachievements.com/PeTeRL90.htm) for helping me trim this solution by clarifying what exactly needs to be done!
All you need to do is use the special attack with everyone that each game starts with. This can be done in each game by pressing X or whichever button you assigned to the special function.
For Streets of Rage simply press X (or whatever you assigned to Special attack) with Axel, Adam, and Blaze. The special attack will result in a police car driving up behind you and shooting a rocket launcher which will then ignite and kill everyone on your screen. You'll be doing it a total of 3 times, since you need to do it once with each character.
For Streets of Rage 2, you need to press X with each character. It doesn't matter if it's a passive or active special for SoR 2 or 3.
For Axel, the passive attack one looks like he forms a flaming dragon on his arm and he does like a windmill punch. His active one is a flurry of rapid punches.
For Blaze, the passive one is a backflip kick and the active one is a hadoken of sorts (similar to Chun Li's kikkuko [sp] attack Street Fighter 2 Turbo.
For Skate, the passive one is a spinning kick on the ground, where he lies down and spins around with his roller blades out. His active one is a "corkscrew kick" where he jumps in the air and literally corkscrew's downward at a forward angle.
For Max, the passive one is a spinning clothesline, similar to what Zangief in street fighter does. His active one is a tackle type of attack where he charges and dives forward.
For Streets of Rage 3, you also need to do each character's special attacks by pressing X, or holding forward and pressing X.
Axel, Skate, and Blaze have the same specials as in Streets of Rage 2, while Zan's passive has him cover himself in electricity similar to Blanka in Street Fighter. Zan's active special attack has him extending one of his arms forward and charging it with electricity.