In each of Double Pug Switch's levels, you can collect two coins - the majority are yellow and three are purple. This is for collecting the yellow coins.
It is important to note that you must actually have the 1000 coins in hand to earn the achievement. As there is an achievement to buy five hats as well, make sure to grab your 1000 coins and then buy the hats.
It's also worth noting that you don't get credit for any coins you collect until you finish the level you're on. You'll get the 1000 well before finishing the game, but if you're having trouble with it, just grind the very first level - you can easily collect around 50 coins in every run as the level has 56 total. You can just pick to restart the level after completing it, so no need to drop to the main menu and reselect the stage over and over.