Perform well enough to unlock the Gold medal on all 12 Arkham City campaigns as Batman. To unlock these maps, you will have to collect Riddler trophies during the main game. See "Genius" for more info on those.
The campaigns are a combination of previous Combat and Predator challenges. They come in sets of three, with three medals to earn in each section for nine total in each. You have the ability to retry up to three times if you fail to get all three medals on a section, because you'll need all nine medals in one run for each.
You are also given a set of mutators that will alter the gameplay which must all be used at some point during the campaign. You can use up to three in a single section so if you feel confident about a section you can use a bunch to get them out of the way early, making a harder one down the line a little easier.
Click here for a complete video guide to these challenges.