Every character has three Heroic Deeds he or she can complete. Two of these "mini-missions" earn Boosts; one earns an additional costume (with the exception of one character who does not have an alternate costume). All three Heroic Deeds earn Dossiers and Concept Art. In order to complete the Deeds you must be controlling the character; or, the character must be joined with you in a Fusion. For example, if you are playing Jean Grey and initiate a Fusion with Iceman, any kills made by the Fusion count for both Iceman and Jean Grey.
Some deeds can only be completed after choosing Pro-Reg or Anti-Reg. If you play a Legendary+ game, you can continue working on Heroic Deeds of characters from your previous game; and, you will unlock Heroic Deeds of other characters if you make a different choice.
For example, in your first game you choose Anti-Reg; but, by the end of the game, you still have not controlled Captain America for 50 kills. You can still get those kills at any time in your Legendary+ game. Further, after choosing Pro-Reg in your Legendary+ game, you can get Iron Man's 50 kills as well.
You can complete Heroic Deeds using Simulator missions or by replaying old maps. This is sometimes the easiest way to complete the Deed. For example, you can get the "Lead team for a whole map" Deed just by replaying an HQ map and immediately going to the "Next Mission" console to end the replay.(This does not work for hero deeds but does work for the complete maps in co-op trophies/achievements)
To check the status of your Heroic Deeds for any character, open the game menu, scroll over to "Stats" and select it. Scroll right once to "Heroes" and scroll down to the hero you want to view. Unfinished Deeds begin with "Defeat" or "Lead". Finished Deeds begin with "Beat" or "Led."