Head downstream until you come to a clearing. Wait until one of the many deer in this area wanders past. If you go slowly and quietly, you'll be less likely to scare them away. Note that Lara can walk with her bow drawn (hold Right Mouse Button/L1 or L2/LT), and this will help her move more stealthily. When a deer passes close by and pauses to drink or listen, draw your bow (hold Right Mouse Button/L1 or L2/LT), zoom in (Z/right stick button)...
...and aim with the mouse or right stick.
Charge up a power shot if possible (briefly hold Left Mouse Button/R1 or R2/RT), then release the Fire button to shoot.
A charged shot to the head, as demonstrated above, will bring down a deer. If the deer doesn't die with one shot, you can either pursue it and try again or attempt to kill a different deer.
When you succeed, approach the fallen animal to trigger a cut scene. Lara uses an arrow to butcher the deer but not before apologizing for having to take its life.
Afterward you're awarded 150 XP, enough to unlock a Skill Point. The on-screen prompt advises you to return to camp.
Head upstream to find the path leading back to the Sheltered Ridge camp. Use Survival Instinct (above) and/or your map (below) if necessary to help find it.