After you have gotten so far into the story you'll be able to build a Memorial and later you will be able to upgrade it to unlock all 23 vigils. Each vigil is essentially a challenge to complete. When you complete a challenge you will recieve an inventory item specific to that challenge. Take the item to the Memorial and select the vigil with a red star over it to "complete" the challenge.
I reccomend completing the vigils in your first playthrough. Once you have completed the game for the first time select "Continue" from the Main Menu to load your game just before the last level. You can then use "Who Knows Where" for the weapon ones as well as the proving grounds.
If you're having trouble with any of the vigils then see this guide for reference (credit to L0cke89 at x360a):http://www.xbox360achievements.org/forum/showthread.php?t=32...
The only thing I would add that is not in the guide... The Proving Ground for the Flaming Bellows is also very easy if you have 3 or 4 upgrades on it. TSG Dwaggie comments that rolling after each of the first two groups of enemies will make things even easier.
The proving grounds I completed (and found the easiest were):
Bow, Repeater, Musket, Hammer, Flamethrower, Mortar, and Pistols.
EDIT: Guys... if you're going to give a negative vote then at least let me know why so I can improve the guide.
Note (Credit: JORAX) : If you start a New Game Plus before completing this achievement, it keeps some vigil records, but not all. For instance, you will have to re-do the Proving Grounds challenges. Just something to keep in mind!
Note 2: (Credit: Zedarboy): "One note about "The Menders" on that guide you linked, they say that the enemy needs to be under 20% hp when you defeat it with a counter, but it's actually YOU that needs to be under 20% hp. After trying for ages I just let a mob hit me until I was under 20% hp, then shield countered, killed it and got The Menders."