The barriers are located in the game mode called drops. There are two drops levels in the game. They are: Wooden Chipper Chase and Lazer Grind Run. You can tell that they are drop levels with the arrow pointing down symbol on the map.
I cant tell you which map to play, since from the look of it, they seem to be mostly the same. Only the visual is different. I however prefer to grind this out on Wooden Chiper Chase.
To break a barrier you need to move the character with the same color as the barrier over the barrier and it will break on its own. While the game make it sound like you should not move the character with a different color over a barrier, there are no penalty for doing so. So I like to move both controllers over the same barrier. The barrier will still break as long as the one with same color is standing in top of it.
But only the barriers with a color count for this. The flassing down arrows do not have any barriers and do not give any points or count towards this achievement. The colorless barriers can not be broken. I did test it out a little bit, and it seems that the colorless barriers are active for 3 and 4 players. So if you have a friend or family member to help you out. Then this will go by much faster.