Age of Empires II: Definitive Edition

Fast and easy way to do it.

A micro management skill check, essentially. Not easy but short and quite straight-forward, because your options are limited.

Some tips:
Play the scenario once, with the map revealed ("marco" + "polo") to get a feel for what you're up against.

Make a save after configuring your unit stances, control groups and research (cav upgrades) & production queues. Now you have a good retry point when you need it.

To supplement the starting army I built triple racks and spammed spearmen for cheap and fast distraction and mixed in some scout cav who you can send out to get the reinforcement units, mainly the monk.

The main thing is getting your monks to convert the enemy War Elephants as soon as you have the mana. You'll fight two sets of two Elephants, then there's two more at the red base just south of the castle.

Make a save when the castle is at half health! The king always flees the castle before it's destroyed. Try to body block possible exits with your units and box him in.

Age of Empires II: Definitive Edition (PC) - Achievement - Chasing the Crown

Kill King Takayutpi before 5 minutes have elapsed in the first Bayinnaung mission "The Burmese Tigers".

This is a relatively difficult timed achievement. Five minutes is not a lot of time to track the King across the map, kill a few units along the way, destroy his Castle and then kill him.
This method is basically just to do all of the above as fast as humanly possible. There are a couple of things to note which should make this a little easier. Firstly, one of your starting heroes is basically a Monk and as you chase the King across the map, you will come across a few enemy elephants. It's important to convert as many of these as possible as you will need the damage to take out a Castle shortly as quick as possible and elephants will be able to soak up a lot of the damage too.
Secondly after you get to the initial camp where the King is and he flees, if you continue right, you will get control of a few more troops including a Monk. With this second monk, make sure to convert elephants on your way to the Kings castle as well.
The third thing to note is that the King will exit the Castle once it reaches a certain level of damage and he will make a run for it. One tip for making this easier is - As you are destroying the castle, send one fast moving unit further to the top right of the map and you will get some ranged horse-riding troops that will make it easier to snipe the King when he does leave the Castle.
Do all of the above as fast as you can and if at first you don't succeed, just keep trying to do it quicker and quicker and shave some time off. Your starting heroes are all expendable, so don't worry about losing them either.

AOE 2 Console Edition

Video recorded Feb 23 so the tricks listed above still work fine.