This can be done two ways:
1. You can start a round and do not get any words. When you finish you will see a leaderboard screen. You can either tap on someones name to add as a frenemies or type in someones name under "Manage your frenemies". Simply play another round after doing so and beat their last score. When the round is over the achievement will unlock.
2. If you have a friend who has played the game within the past 24 hour, you can simply try and pass their score without adding anyone as a frenemies.
This is what the "Frenemie" page looks like. When selected, their names are a lighter color than the background
Image provided by StayonTarget
WARNING: Use the Wordament Solver at your own risk. The developers haunt the leaderboards for anything suspicious. They will ban you if they suspect you have cheated and you will then no longer be able to get the achievements.
Wordament Solver - http://dev.360surface.net/Wordament/EnterBoard.aspx