Middle-earth: Shadow of Mordor - Game of the Year Edition

There are 10 Azkâr Legend missions required for this achievement, all of which are available in the primary Story mode. They are very straightforward and come down to "shoot what the game tells you to."

You can also do these whenever you'd like for the most part. A few require certain upgrades to be unlocked, but what I meant was that you can do these before you play the story or after you beat the game. I don't think it upgrades your bow in any way other than aesthetically, so it does not matter when you finish them all.

However, be aware that they give a lot of XP per mission, which could unlock useful upgrades that do make the game easier.

Also, to make these considerably easier you should go into them with a few Runes. I used the following Runes, which made the bow so overpowered that there is barely a reason to use anything else in the game:

Ascendant (Epic) - Reduces focus depletion by 50% (comes with the GOTY version).
Deadly Archer (Epic) - Increases headshot damage by 25% (comes with the GOTY version).
The following runes can be any level, but higher levels are better:
Keen Eyed - Recover focus for stealth kills with the bow
Broken Mind - Chance to recover Elf-Shot with Drains or Brands
Sylvan Arrow - Chance to recover Focus for a headshot kill

Combined with upgraded Focus time and more Elf-Shot, I could clear out entire camps without leaving Focus mode, and would only really need to leave Focus mode because I ran out of arrows. The bow challenges will be a breeze with all of this equipped.

Here are the challenges you will need to complete. Credit for this list goes to Zasta 360GameTV for thinking to transcribe all of the information before completing the achievement unlike me.