The achievement's description is wrong, you need to do 50 legendary quests, not 25. Legendary quests unlock once you've finished the story (chapter 9 is the last). You can check how many you have finished by pressing back and selecting inventory, then pressing the left stick in to see your stats. Scroll down to the bottom to find how many you've done. Note that the count shown is only for that hero, you have to check the stats of each hero to get a total. However, the achievement unlocks when you've finished 50 total across all heroes, I had 17 with Black Widow and 33 with Jean Grey when this unlocked.
Once you've finished the story, you'll have an epilogue that will walk you through a few types of legendary quests. A sample of what it may ask you to do.
-Kill enemies/bosses/objects in a story chapter or operation
-Enter a boss challenge room and kill the boss.
-Finish a danger room scenario.
-Enter midtown patrol and kill enemies/bosses/objects.
-Complete a certain alert in a story chapter.
The game usually highlights objectives when you get close to them. Sadly, some of them are lightning quick and some are annoying as hell and the amount of legendary marks doesn't always match up with effort required. There doesn't appear to be a way to reroll the quest, so if it's one of the sucky ones, not much you can do but power through.
The main point of doing these is getting legendary marks, which you can take to Cleo to purchase legendaries. You need to level a legendary to max for an achievement. 50 quests is more than enough to get the Helm of the All-Father if all you care about is the achievement. If you want any of the other ones you need 300. You get 1-3 from each legendary quest on normal difficulty.
Eisenhovver commented that he gets 4-6 from cosmic difficulty legendary quests, so if you can handle the bump in difficulty, that can help you get to 300 much faster.
EDIT : Sadly some of the quests are bugged, so I'm going to write notes here about them to help people who have the same problems.
- Had troubles with the Pull the Plug quest, where you need to kill 25 AIM engineers, and found this thread :
https://forums.marvelheroes.com/discussion/318692/legendary-.... It seems the person who stated that only specialists having conversations count for the 25 AIM engineers is correct, as that's been my experience through multiple playthroughs of the quest.
-For Send 'Em Packing, saving civilians doesn't move the counter. I only got credit while doing the alert in the northeast part of the map. You'll have to do the alert twice to get all 3. Pretty annoying.