This one is very easy, unless you make it harder for yourself.
First, do this on Pilgrim difficulty, as the difficulty level does not affect the achievement. You will not need to eat and drink as frequently, and cold weather conditions will take longer to affect you. Plus, you should start out with better gear.
Start in any region you prefer. I did this in Mystery Lake, as shelter and resources are plentiful there. No matter where you spawn, look for these things:
-SHELTER. Find a building with a protected interior as soon as possible. Many interiors have fireplaces, which you will need.
-FIREWOOD. Fires are the lifeblood of The Long Dark, and firewood is in turn their lifeblood. With a fire (indoor or outdoor), you can cook meat and other food to prevent food borne illness, and at most locations you can melt snow and boil the water to produce potable drinking water.
Once you have shelter and firewood, you are set for heat and water (at least for as long as you have fuel for the fire). At this point, you just need to find some food to last you through the day.
Your body will consume about 1,200 calories in a 24 hour period (depending on how active you are and how much weight you carry). If you can find a deer carcass, and you have a method of stripping it, you should be golden. In most cases, though, you'll probably have to scrounge to find abandoned snacks.
If you have enough food, water, and a warm shelter, you can simply sleep or wait until you get the achievement. The achievement will pop when you have survived for 24 hours.