Age of Empires II: Definitive Edition

First off put your Hero unit in the nearby purple castle and forget about him.

Make sure to not destroy the 3rd castle before killing all the Turkish factions. Just ignoring them and killing the castles won't earn this achievement as it doesn't fully satisfy all parameters as read from the description.

They will keep making units as long as they have military buildings. It's trigger base, killing all their houses won't do much. They also won't restore anything you destroy but you'll have to hunt their civilians to get them to resign.

You'll keep getting reinforcements from your starting position. Speaking off, you can collect some units in the beginning, such as the 4 cav archers in the east corner.

Since they'll spawn infinitely, make use of your 175 pop cap. Make 50 or so mixed cav units (back them up with cav archers) and put a handful or more monks nearby. Then pump out some Bombard Cannons / siege units.

Together you can wreck all military buildings but block all units from reaching them.

For the Byzantine castles, I recommend Cannon Galleons, they can reach all 3. Make sure to have a few Fire Ships with them to guard against the War Galleons.
Green can perhaps also be demolished by sea but I didn't try that.

As for eco, everything is pretty scarce until you go out of the middle. I recommend getting wood in the south near purple's defense buildings. Good luck. Make sure to use 10 cav units for base defense.

Age of Empires II: Definitive Edition (PC) - Achievement - Turkish Delight

Win the Historical Battle "Bapheus (1302)" by defeating all other Turkish factions without forming an alliance with any of them.

Apologies in advance if your following the video - This one is quite long winded. The reason being is that you have 3 Turkish factions to deal with prior to taking on your main enemy. The Turkish factions must be dealt with first. You are given the option to ally with one of them, but you must refrain from doing so in order to obtain this achievement, meaning you'll be surrounded from almost every side and under fairly regular attack. You'll get regular reinforcements arriving to help you and you also have access to Bombard Cannons which will be helpful as all of your enemies are in fortified positions with walls, towers and castles. Stone is in somewhat short supply, so building your own castles to push forward and claim land might not be an option for you, but once you have your resources in order which your given time to do in the beginning, raising an army with some Bombard Cannons to go with it isn't that hard - It's more just a long process eliminating your enemies given they are all in good defensive positions.