Do not wait to get this until the end. If you have no friends with the game, find some before going to beat the game, because if you set scores for all the missions, and they end up posting before any of your friends' scores do, you will have to clear your game progress and begin anew.
Make sure someone on your friends list has scores registered on the leaderboard. If you send me a friend request, with my ps3 list being full all the time, give me a specific date and time of when to accept your FR, and I'll do so, leaving you a 5 or so minute window to be able to beat any of my scores. If my friend's list is full, shoot me a PM on PSN and we'll arrange a date and time to coordinate so that I can make room on my friend's list for a short time.
Once that's done, check the leaderboard for each mission and find one where a friend as scored low. Simply get 1 point higher than their posted leaderboard score for that mission, and this trophy will be yours.