Alright, so I had this typed out and ready to post but saw that KungFuPanda17 already had a solution. Being that it took forever to type, I decided to post it anyway. Warning, contains spoilers.
So after the cut scene with your family, you are down on the final platform with the Baron. He is on top and you are down below you have to use the dolls and their available abilities to beat him.
So first stack into your father, William Blackmore, then Pump the Bellows into the chimney to start the fire then either use the doll with the Oil Can or Meriweather Malodor (Flatulence Doll) to make the fire roar, burning the Baron.
Then, stack into Jasper (a maintenance doll) and use him to open the hatch, then stack into your mother Agnes Blackmore and use her clean out ability to knock off the Baron.
Stack into The Widow Chastity, then into your brother Archibald Blackmore, use his chimney climb ability to get to the top, unstack to Widow Chastity and seduce the Baron off the ledge.
Stack into your sister Abigail Blackmore snuff out the 3 little fires blocking the gate then stack into Cromwell the Terrible and use his Royal Wedgie ability to open the gate, knocking the Baron off.
Stack into your sister Agatha Blackmore and use her parasol spin to block the falling coal, then stack into Ursela Allendorf (German Woman from the Train Station) and use her ability to scream into the pipe, startling the Baron and making him fall.
Finally, stack into your brother Albert Blackmore then stack into the pelican, fly to the nest using the pelican, unstack to Albert & use his shovel to knock off the coal onto the Baron.
After all this there's a cut scene, and you'll be behind a gate with Charlie, the Flare Doll and Meriweather Malodor, stack into all the dolls, walk to the gate use the flare with the flare doll and then unstack to Malodor and pass gas, boom, gate explodes and another cutscene.
Now you are down below and playing a final game of rock paper scissors with the Baron. Just watch and see what one he is using then stack into the one beating that.
Rock smashes Scissors, Scissors cut Paper, Paper covers Rock.
Rock Doll is Blue.
Scissor Doll is Yellow.
Paper Doll is Red.
Also I noticed if you can hit him while he is running to another doll, it counts. Do this 4 times, cut scene, then beep-boop, achievement.