1st Step: You may find some NPCs talking about Altius, Shield Knight's pet owl, whom you'll find on the sign next to the Well's entrance.
use your shovel to hit him, after that he starts flying around. Shovel drop on him when he's directly above the Well. Dropping him low enough will make him fly downward into the Well. Doing this once will make him automatically go down anytime you start a new run.

2nd Step: Find him in one of the Mushroom Mine's side areas, where he will be standing on a chest. Hit him to make him get off the chest and give you some gold.
After beating the first boss, you will meet up with Shield Knight AND Altius.

3rd Step: You can either do this in Smeltworks or Secret Fountain first. You must find Chester in one of the side areas. Altius will be flying above the chest. Chester will have an item called the "Cog on a String." Buy it (assuming you have enough gold), and ALSO remember to collect the three gears in the first level of the world you're currently in. When you reach the first checkpoint, assuming you have all three gears and the Cog on a String, Altius will jam it into the Drill Capsule, preventing you from using it for the duration of your time spent in that world.
Once you beat the boss, you will meet up with Shield Knight and Altius again.

4th Step: You can either do this in Smeltworks or Secret Fountain, depending on what area you did first. In one of the side areas of each level, you will come across a Griffoth Statue. You will need to sacrifice one of your passive items to unlock the doors, allowing Shield Knight to drop down them. Upon unlocking all three shrines, and beating the boss, you will meet up with Shield Knight.
Talk with her and she will gift you the Skeleton Key.

5th Step: You can do this in either Magic Landfill or Grub Pit first. Take the key to a skull door that matches its color.
Upon unlocking it, you will find the Omega Driver, which is necessary for dismantling Drill Knight's tech. Altius will hold onto it until you reach Drill Knight's Castle.
After beating the boss, you will meet up with Shield Knight again.

6th Step: You can either do this in Magic Landfill or Grub Pit, depending on what area you did first. Shield Knight will give you a bomb that must be placed in front of an explosive wall. Blowing it up before the bomb goes off will lead you to a room with a pipe that leads to Drill Knight's Castle. Altius will take the Driver down with him.

7th Step: Once you reach Drill Knight's castle, Altius will give you the Omega Driver. DO NOT use the Omega Driver or trade it out for another relic.
Once you reach Drill Knight, fight him as normal, but once he's at three health, he will activate the Omega Saw. Use the Omega Driver to destroy the Saw, and it will trigger a new cutscene.
NOTE: You must throw it when the Saw on the same vertical position as you. Shovel Knight will throw the Driver at the Saw in a horizontal line.

SPOILERS ! ! !
You will have unlocked a new area, the Crystal Core. Beat it and you will unlock the achievement.