There are two ways you go for this achievement, but no matter what, it can only be done in single player. The first way you can do this is by play the game normally and remove only middle pieces, and you can get 1000 points that way. Make sure to not let the tower lean too far to one side by placing the blocks gradually leaning to the other direction. It is a little tricky but can be done. Be sure to watch out for wobble as this causes you to lose points. As soon as you get 1000 points, knock the tower down quickly and you will not lose any points for wobble and unlock the achievement.
The other way is a much faster, but it may take a few tries to do. When the game starts, find the block facing away from Mr Potato Head about 3 rows up. Grab this block, and pull it out and to the side. The trick here is to make the tower turn about 90 degrees then wobble before falling over. If you knock over Mr Potato Head, it will not count. Make sure you topple the tower towards the screen(the side of the table to Mr Potato Head's right).
The tower must wobble, be turned at least 90 degrees, and cannot knock over Mr. Potato Head when it falls for this to work. For some reason, the game gives you some insane score and even after the wobble penalty, you will have more than enough points. This can be tough to pull off but may be easier for some people compared to playing a full game to get the 1000 points.