Tomb Raider

Bow Location: You receive the Bow as part of the story by claiming it from a hanging corpse after your first night at a camp fire.

Firing: To fire an arrow you simply hold LT to aim, move the reticle until it is over an enemy, at which point it will turn red, then pull RT to fire.

Greater Damage: If you hold RT you will see the reticle throb as it closes in on itself and becomes a red circle. This means that when you release RT the fired arrow will now do more damage.

Getting Kills when in Stealth: You'll get a lot of opportunities to sneak up on people, aim, move the reticle over their head, hold RT until the increased damage reticle appears, release RT and voila, a one hit headshot kill with the Bow.

Getting Kills when in Mid-Range Combat: A lot of the time I found that the AI did the classic 'bobbing' behaviour - i.e. sitting behind a crate or a wall and helpfully popping its head out from time to time for me to shoot at, so that was just a case of patience.

Getting Kills in Close Range Combat: Sometimes Melee attackers rush you and if you can't get the head shot it will take 2 or 3 arrows to bring them down. In these cases I pretty much always used the Pistol or Shotgun (contributing to other achievements) as there are more than enough opportunities for Stealth and Mid-Range Combat kills with the Bow in the game.

Usefull Skills
There are always plenty of arrows lying around, but if you do find yourself struggling for arrows unlock the 'Arrrow Retrieval' skill as soon as possible. This allows you to retrieve arrows from killed, unarmoured enemies.

The bow is the first weapon you acquire in the game. It is story related and cannot be missed. You must kill 50 enemies with it. To make it easier, I recommend only killing enemies with the bow until the achievement unlocks and playing on Casual difficulty. I found that most enemies went down in 1-2 shots on Casual. Try to go for headshots simultaneously as there is another achievement for that.