When playing Uno with standard rules, if at any point it is your turn and you cannot put any of your cards down you must draw from the deck. To get this achievement, you must play a game without ever having done that.
Your best bet here is to play on Elimination, as the game is faster and card drawing is unavailable anyway. Win an Elimination game and you should have the achievement in the bag.
You can't expect to get this in online play, other players will always force you to pick up some cards eventually and matchmaking will just make the process take longer, so stick to single player.
So many of the achievements in this game overlap that while going for this you will likely get some others and visa versa.