Princess and the Pauper

If Princess does not follow you, talk to Princess one square west of Rodney.

Talk to Aster at Fort Forinthry

Talk to Overseer Siv in the Command Centre

Exit the Command Centre and stand west of it, just by the wooden barrels.

Talk to Princess to obtain a lost sausage.

Talk to the Cook in the Jolly Boar Inn.

Head north to the Wilderness wall just outside the inn and talk to Princess to obtain a half-eaten meat pie.

Go to the small cabbage patch south of the Blue Moon Inn within Varrock and talk to Princess.

After the cutscene, talk to Princess again. Pick up the raw beef nearby if you don't have one.

Get the ingredients for the dinner:

Use a hammer on raw meat to obtain tenderised meat.

Pick the redberry bush south of Varrock, near the elder tree. Do this after the conversation with Princess, otherwise you won't get the juiciest redberries.

With a free inventory space, talk to Gertrude in her house about the secret ingredient.

If you have started Rat Catchers,

Use the raw salmon on the range in Gertrude's house to make a gently smoked salmon.

Use a bowl of water on any of the ingredients to make an uncooked dog's dinner.

Cook the uncooked dog's dinner on the range to obtain an unseasoned dog's dinner.

Use the finest doogle leaves on it to make a perfect dog's dinner.

Talk to Princess to show her the completed perfect dog's dinner.

Go back to the cabbage patch near the south entrance of Varrock and talk to Princess there. She will lead you back in the direction of the Jolly Boar Inn.

Go back to the Jolly Boar Inn. Stand inside, west of the kitchen, and talk to Princess.

After the cutscene, talk to Princess.

Go directly north, and talk to the stray dog outside the inn (the dog is just south of the wilderness wall).

Talk to Rodney at Fort Forinthry.

Talk to Bill.

Talk to Bill again with the materials in the backpack.

Choose a name for the dog to finish the quest.