Far Cry 2

For people just starting and wanting to get this ASAP. Save before attempting so you don't lose the buddy.

If you are playing as Marty (the default character) the first opportunity to do this is during the tutorial when visiting Mike's bar for the first time and talking to Flora Guillen (standing against left wall next to table when you walk in). This buddy will instantly become "rescue-ready" after completing the conversation, which you can verify at the Pause menu (picture in lower-left of screen with buddy and status listed below). I haven't tested or researched storyline impact with a different characters since this is my first (and only!!!) playthrough.

-With your buddy rescue-ready, find an enemy (or group of enemies) and take damage until all your health is gone and the screen goes black, initiating the buddy rescue cut-scene. If you are using the above advice and doing this at the first chance, take a right out of Mike's bar and drive shortly NorthEast along the road to the Lumber zone for a quick enemy encounter.

-After the buddy rescues you (earning the "Out of the frying pan..." achievement if new to the game) you only need a single machete swipe to your buddy to mortally wound them. I do not recommend shooting in the legs because I killed my buddy a few times doing so. It is safer to clear the immediate area of enemies before sadistically harming your buddy for achievements, since I didn't and found myself getting shot at.

-Once you've mortally wounded them with a single machete swipe (or your own choice of damage) they will hunch over in pain and after a few seconds of delay (don't hurt them anymore or they will die and you won't get a button prompt!) a Y button prompt will show on screen. Again, if following my advice and doing this for the first time, you will have a few tutorial screens to skip through that explain your choices to kill (by pressing left or right on the d-pad to draw your gun) or heal (Right Trigger with syrette out by default) your buddy.

Note: You can do this achievement and "...into the fire." (Save a Buddy from a mortal wound) at the same time. Simply heal them with the syrette (RT) and hit them again with the machete.

-Press left or right on the d-pad to change to your gun, press Right Trigger to shoot your buddy, ending their suffering and popping the achievement. You can instantly reload your previous save to avoid losing this buddy permanently.

Another long and verbose solution for an easy achievement from yours truly. Apologies to anyone further in the storyline using this solution for all the extraneous information. I typed this from scratch after using the existing solutions and wasting a few attempts. Credit to Dog of Thunder for the machete-swing method, very sure-fire.

If you find anything in this solution lacking, please take a moment out of your busy schedule to comment and allow me an opportunity to improve BEFORE voting negatively. Happy boosting!