There are 30 posters in the base game that need to be tagged.
Ten are Harvey Dent posters in Gotham City that you can spray in free roam:
Characters that can spray-paint are Joker, Riddler, Two-Face, or any variation of Harley Quinn.
Completing these 10 will also get you the gold brick for the Poster Painter challenge.
The remaining 20 hero posters are in each of the 20 story levels, some can be tagged in the story playthrough but most will need to be done in free play.
This video series will guide you through all collectibles for each of the 20 levels, including the posters:
Credit for the videos to HappyThumbsGaming.
There are also 6 posters in the DLC levels, these shouldn't be required for the achievement but I couldn't test because I already had all of them sprayed before I completed the base game posters.