Home Sweet Home! Achievement in Little Kitty, Big City

Story-related; cannot be missed.

This is one of the only story achievements in the game, and it's for beating the game. Don't worry - you can still free roam after beating the game.

This is the main goal of the game and will require you to collect all of the fish for Fin-tastic! (25G). Once you have done that, return to where you trade Shinies with the Crow to earn your first fish. With a full stamina bar, you can begin climbing the vines on the skyscraper building and continue progressing to your window in the center of the map. A quick summary of how to do this is in the roadmap, but to sum it up:
 

1. Get out of the starting area and talk to the Crow, who asks for 25 Shinies
2. You can grind out infinite Shinies from recycling cans. Refer to Fat Cat (25G) for how to do it quickly and easily
3. The Crow will then give you your first fish
4. You will need to open up the rest of the city now by teleporting with Tanuki by the construction site, which requires three feathers from catching three birds. From the construction site you can move crates to access more. You can also approach those dogs blocking your path from behind so that you can put bones in their dishes to remove them
5. The other three fish you need are as follows:
   1. In the supermarket in the south center of the map, you'll need to drag the fan out of the way so you can get to it
   2. At the waterfront in the northwest corner of the map, wait for the fisherman to catch a fish, and when it lands on the ground by him, eat it
   3. Near the center of the map is a building you can climb (only needs one fish to reach the top, so this doesn't need to be your last one) that has a woman spraying water on her garden. Go past her on the right side to reach the shutoff valve for the hose. Leave through the door she opens and jump in her window to find the last fish

Once you've eaten all four fish, you can head to the tallest building in the center of the map and climb to the top now with max stamina to reach the window. A cutscene will play and you'll get this achievement.

Here is a full video walkthrough (timestamps in YouTube description):