***UPDATE***
This game, and therefore all of its achievements, have been discontinued.
***OLD SOLUTION***
During some story mode maps, there are yellow doorways with an ! instead of being a treasure room. These are supervillain showdowns, and once you complete seven different ones, the achievement will pop.
They are each located on these maps - based on comments, each spoiler tag below the location will have some relative directions to each door:
Chapter 1 - Hell's Kitchen South - Electro
*** Spoiler - click to reveal ***Before the building with Venom, along the linear path, it'll be inside the fenced-in space just before the cop cars but after the church and park
Chapter 2 - Shipping Yard - Rhino
*** Spoiler - click to reveal ***This one's also hard to miss, along the main path. It's between the area where the two treasure rooms are, and where the shipping containers change color from blue to yellow.
Chapter 3 - Bamboo Forest - Gorgon
*** Spoiler - click to reveal ***Immediately when you start at the waypoint, use your travel power to fly along the upper/outer edge of the map (from the top border above the waypoint, you'll go up and to the left initially) - when you get to the Sacred Dojo treasure room, Gorgon Falls will be directly east/right from there, or you can keep following the edge you've been following
Chapter 3 - Lowtown - Viper
*** Spoiler - click to reveal ***Also along the main path - start out going up and to the right, and the path will naturally curve down a bit, the back up-right again - stay to the left at the small fork and you'll see the Hidden Lounge treasure room - follow that wall around and you'll go up and slightly left, past the stairs you could climb, and up ahead you'll see a building with a lit sign that says "Princess Bar"
Chapter 4 - Lower East Side - Green Goblin
*** Spoiler - click to reveal ***The easiest way to get to this one is actually to use the Upper East Side waypoint,
and then use the blue subway portal next to where you spawn to go back to the Lower East Side... from there, go directly away from the portal (as if you were walking out of the subway stairs and straight ahead once you did so) and go around the Theater building. The Showdown is in a building called Madame Erika's House of Wax.
Chapter 5 - Wretched Slum - Lady Deathstrike
*** Spoiler - click to reveal ***This is possibly the easiest one to find. Go directly south/down from the waypoint and you should see it pop up on the mini-map - it's about a block away from the starting waypoint.
Chapter 6 - Hunting Grounds - Blob & Toad
*** Spoiler - click to reveal ***Once you exit the building that houses the starting waypoint, follow that upper level border with your travel power, going clockwise/right/east-ish. You'll pass two treasure rooms and the Deep Tunnels portal, and the Showdown is just past this to the east/southeast,
and the entry is for a Circus area.
Chapter 7 - Mutant Marsh - Sabertooth
*** Spoiler - click to reveal ***From the starting waypoint, go up-left/northwest to go into the marsh, and follow the bottom wall clockwise (turning left once you get to the marsh from the waypoint) - when you get to the southernmost/bottom point of the map, you should see the Evolution Facility waypoint pop up north of you. Go to it, and then do northeast/up-right from it. The way point is about as far away from the outer wall on the bottom as the Evolution Facility was, about halfway between that waypoint and where you came out from the starting waypoint.
UPDATE: With the current Daily Bugle event / operation going on, if you haven't previously defeated these enemies, they count towards the achievement for defeating ten different Sinister Six villains from the DLC