Once all characters are playable, and Laverne has put on the costume, you will need to use the Fork on the Cat as Laverne (kid with the crazy eye).
The Fork can be picked up by Bernard (kid with glasses in the present) from the Break room. From the Main Lobby where you start the game, open/go through the double doors underneath the WELCOME banner. Then, go through the Swinging door (no need to open, just click on it) located just to the left of the fireplace.
Once in the "Break" room, pick up the Fork on the table in front of you. Then, open your inventory, click on the Fork, then click on Laverne to automatically put it in their inventory (the game automatically sends both players to the Chron-O-John to flush the object to each other).
To get the Cat, you will need the Booboo-B-Gone and the Squeaky Mouse Toy.
The Booboo-B-Gone can be picked up by Bernard. From the Main Lobby where you start the game, open/go through the door with the OFFICE sign on it behind the counter and left of the grandfather clock. Close this door again for an achievement if you haven't gotten it already. Once in the office, open the desk drawer to reveal the Booboo-B-Gone. Pick it up, open your inventory, click on the Booboo-B-Gone, and click on Laverne to automatically put it in their inventory.
The Squeaky Mouse Toy can be picked up by Hoagie (portly kid with the backwards cap). Go to the attic of the Inn. Once there, you will see a cat (not THE cat unfortunately) sitting on the floor playing with the Squeaky Mouse Toy. First, go "use" the squeaky mattress to the left-front of the room. Then, "use" that bed with the bed in the back. You should see a cutscene where Hoagie struggles to pick up the mattress and a sculptor comments on the loud noises in the attic. Once the beds are switched, "use" the back mattress (now the squeaky mattress) and notice that the cat is interested. As the cat approaches the bed to investigate, quickly pick up the Squeaky Mouse Toy on the ground in the center-front of the room. Open your inventory, and send the toy to Laverne.
Now that Laverne has the Fork, Booboo-B-Gone, and Squeaky Mouse Toy, have her head outside and go right until you see a Cat (THE cat!) struggling to get under a fence. Use the Booboo-B-Gone on the fence ("this outta be good") and observe the Cat go under the fence, notice that it has a white line down it's back and tail, freak out, and head to the roof. Now, while Laverne is on the ground and the Cat is on the roof, use the Squeaky Mouse Toy on the Cat to have it dropped into Laverne's inventory.
Now, open Laverne's inventory, select the Fork, and use it on the Cat.
Laverne will say "He's not even cooked." as your achievement pops.
If you see a way to improve this guide or want further clarification, leave a comment! Thanks!