Age of Empires II: Definitive Edition

This mission has you start off in the north of the map with a small force. In the middle of the Italian landmass you're on is the red Normans; to the west is the yellow Lombards, and to the far south and east the magenta Byzantines. To the far southwest on some islands are Saracen pirates (green). To get the achievement, you don't ally with either the yellow or magenta forces; instead you need to destroy one of the faction's castles.

First things first, run down to the first Norman base to the south, surround the forces there, and then change your diplomatic stance to enemy to kill off the soldiers and capture the area for yourself. Keep adding and upgrading units and take the next two donjons.

Now, you can either take on the Byzantines or the Lombards at this point, but I chose the Lombards (yellow guys to the west), since their castle is *much* easier to access. Capture the southernmost Lombard camp, and then have your troops hug the east side of the path and go past the camp with the castle in it, avoiding it entirely. Instead take the camp with the town center. From there, you can get your economy humming. You'll need 2 or 3 rams, and having a few priests can help too. Occasionally you'll receive forces by boat you can land on the beaches. You can if need be destroy the two other camps north of you, but I didn't need the population; also, the saracens (green) will likely threaten your southern camps, but honestly you can sort of ignore them if you want once you repulse their initial attacks, since we won't have to worry about the game going long. Just bash down the gates of the remaining Lombard base, kill everyone in it, and clear a path for your rams to the castle. Once it's down, you've won.

Age of Empires II: Definitive Edition (PC) - Achievement - Self-Made Man

Win without allying with Argyrus or Guaimar in the mission "Guiscard Arrives".

You have the option to ally with Argyrus the Byzantine (Purple) or Guaimar of Lombard (Yellow) by sending your hero to either of their castles. To obtain this achievement, you must refrain from either and destroy one of their castles.
I opted for attacking Guaimar as this seemed easier based on the location of his towns and Castle. You can take a couple of settlements with your starting army. Once capturing the settlement closest to his Castle, I built 5 Battering Rams to breach the gates and take down the Castle, whilst using my main force to protect them and kill the units in the town.
Once you destroy the yellow Castle, the hard part is over and you can build your forces up and take the remaining Norman Barons (Red) that you need to achieve victory without ever engaging Argyrus.
As you get reinforcements and resources with each settlement you take, you shouldn't need many villagers to maintain your resources, so build a few to begin with if needed, but remember to kill them later so you can use the population to support a larger army.
You will also periodically receive reinforcements via Transport Ship from the sea.