Neighborhood Hero Achievement in Little Kitty, Big City

There are 24 quests in the game, but note that only the ones on the leftmost tab with the checkmark count (the ones on the trophy tab are just achievement-related tasks, and those aren't quests needed for this achievement). All of your quests will come from the various NPCs (animals) around the city. Many are very short and easy, some are a bit longer. Plus, you can back track from returning home to complete any quests you missed.

Here is a list of all quests in the order I found them (this will vary depending on how you explore the city) and how to complete them:
 

1. Become an artist - right at the start of the game, knock over paint cans, walk in the paint, then walk on the canvas. Repeat with each paint can until the quest completes
2. Find that Crow - story related
3. Catch a Bird - story related, Crow will teach you how to do this
4. Help the Mayor get some sleep - at grandma's market (marked by a carrot in the center of your map) is a cat who wants you to knock over jam jars. There are three inside this little market area. Climb up and swat them off to break them all
5. Rescue ??? from the pipe - you'll find Tanuki stuck in a pipe near the construction site. Drag him out to save him
6. Fetch 3 feathers for Tanuki - from the same animal, catch three birds to get three feathers, then return to him to open his first portal
7. Replace Beetle's phone - refer to Snap Happy! (25G) 
8. Bring Crow 25 Shinies - refer to Snack Time! (25G) 
9. Find Chameleon Part 1 - refer to the video below
10. Find Chameleon Part 2 - refer to the video below
11. Find Chameleon Part 3 - refer to the video below
12. Find Chameleon Part 4 - refer to the video below
13. Find Chameleon Part 5 - refer to the video below
14. Find Chameleon Part 6 - refer to the video below
15. Find Chameleon Part 7 - refer to the video below
16. Find Chameleon Part 8 - refer to the video below
17. Reuinte the Family - refer to Quack Troops! (25G) 
18. Steal the fisherman's fish - refer to Fin-tastic! (25G) 
19. Steal the gardener's lunch - refer to Fin-tastic! (25G) 
20. Get back home - this is for beating the game. Refer to Home Sweet Home! (50G) 
21. Boss Cat vs Ramune - in the south central part of the map is a toy store. Head inside and talk to the cat, who will task you with catching a yellow bird. This bird flies around right outside and periodically lands in the street. Catch it like any other bird to complete this quest
22. Pose for Beetle - this one is located up top on a balcony near a hat. You'll find Beetle locked out of a door and asking you to pose. Simply stand anywhere (such as the workout mat) and interact with Beetle to get your picture taken
23. Fetch the dog's balls - this will pop up automatically near one of Tanuki's portals. A dog will stick its head from a bush and ask you to help. It tasks you with finding its three favorite balls, which are all right in its yard in various places (up on a tree, under the bucket near the dog, etc.). There is a also a bird nest in this area
24. Waiting on a sunbeam - this is the "final" napping spot, located high up on a rooftop, which requires you to bring a bagel from a human up top, put it in the food dish, then ring the bell. Refer to Cat Napper (25G), since this quest is required to be completed to access this final napping spot

Here is the video showing the Chameleon's eight locations: