The bow is the first weapon you get. Ammo shouldn't be too much of an issue, especially considering when you fast travel, the ammo respawns. There is also a skill that allows you to gather the arrows back off the bodies of dead enemies. You should try to stack this with unaware kills and headshots. Getting kills resulting in explosions (like fire arrows in oil) do not count. The arrow must be the shot that is the killing blow. The bow is in the slot and you can choose your arrow type there when you get the option.
Note: You cannot reload checkpoints to gain extra kills towards your stats. These must all be done on unique enemies.