Here is the full list of all 10 Quiplash achievements. It takes around 0-1 hour to unlock all of the achievements on Windows.
Win all battles in Rounds 1 and 2
Write "meatloaf candy" and win a battle with it
Enter the same thing as your opponent
Get 70% or more of the votes in The Last Lash
Write a quip that Schmitty actually responds to
Win both battles in a round
Get a Quiplash! (sweep votes in a 5+-player game)
Play a game with an Audience
Successfully enter both your quips in a round
Play an 8-player game