Finding the Dog Shrine:The Dog Shrine is available after you spare Papyrus in his boss fight, and it's located in his house (he'll take you there once you talk to him after his fight). Once in his house, go through his kitchen to find a doorway leading you to the shrine. The box on the left is where you donate 25G for 5 coins, so it'll cost you a total of 2500G to get this achievement. Because it's so expensive, it's recommended you save these achievements for last. By that point you should have a decent amount of cash and 8 inventory slots which will make this next part go MUCH faster.
Making Quick/Easy Money:If you're running low on funds, I found a stress-free way to farm money towards the end of the game. Go back to the waterfall area via the ferry and head right where you'll come across a shopkeeper. Buy 8 CloudyGlasses from him for 30G each. You'll then go right, going back into the mushroom puzzle. The bottom right of this area is the location of Temmie Village, a secret location where the shopkeeper there will then buy these glasses off of you for 50G each, netting you a 160G profit for only a minute's run (video provided below if you prefer to visually see how to get there). Credit to
BlackJay for pointing out you can use the storage boxes next to each shop keeper to store and retrieve more CloudyGlasses per trip.
Once back at the Shrine with 2500G in hand, just exchange it for 500 dog coins, and then put them all in the box next to that one. You'll unlock all the coin achievements at once.
Video provided by YouTuber Luxac. Please note at 0:22 in the video is where the shopkeeper is who sells the glasses (and going down from there is a ferry shortcut to Snowdin where you find the Shrine). Just take the glasses to the village he goes to in the video.