*Requires network connection* When you select a world and are looking at the level selection screen (the one with dots and shows how many stars you have on all the levels), floating above each level will be a picture the player on your friends list that earned the highest score on that level, with a crown above their picture. Essentially you need to work to have your picture there on all 15 of the levels in any world, meaning you need to have the highest score out of all your friends on each level of a world.
You can either go crazy and try to earn all of the high scores legitimately, or more easily, remove any friends from your list that have higher or unreasonable scores to beat. Feel free to write down all the friends you remove if you wish to add them afterwards. I personally just went to the leaderboards screen and removed all my friends that had more "Monsters Found" than me. It took a several minutes for the game to actually recognize I booted everyone from my list. You then just need to go to a world where you have the high score on all 15 levels and the achievement will pop when you enter the level selection screen within the world. If some of your other friends still have higher scores than you, either beat their score or boot them from your list. Repeat until the achievement pops.