This solution can also be used for the achievement Night Walker.
Start a new Survival Mode game on Pilgrim difficulty (you can do this on a higher difficulty if you want a challenge, but difficulty does not affect the achievement). Set your starting region as Mystery Lake.
Now you'll want to make your way to the lake itself. Depending on where you spawn, this may take a day or more, but use the sun to navigate in a southwesterly direction. When you reach the lake, set up a base in the Camp Office building (west side of the lake w/ Canadian flag outside). Take a day or two to set up operations here.
When you're done scrounging through the Camp Office, you should have a few fishing hooks and wires. Use the Crafting menu to make however many fishing kits as you can. Leave the office and head out on the lake. There are four Ice Fishing Huts on it, and you'll want to search them all. In the process, you should find a hatchet and an Ice Fishing research book.
To complete the achievement you have to spend three consecutive nights outdoors. To be on the safe side, you'll want to be outdoors for the entire time the moon icon is visible on the time-of-day display. Also, your character will usually make a comment about surviving through the night.
Use any of the four Ice Fishing Huts as your place to weather three nights outdoors. The Huts will provide you with shelter from the elements (while not counting as "indoors"), plus they have both a pot belly stove and an ice fishing hole.
Using the daylight hours to scrounge for any firewood you might need is the only task you might need to do in between stints of sleeping, waiting, and fishing. As long as you have a fire going during the nighttime hours, you won't freeze to death, and you should also use it to replenish your water supply. Fish as needed; you only need one or two fish to eat per day, and if you read the skill book, that should not be an issue.
Now, it's just a matter of time. Sleep for as many hours as you can during the night, and if you have the firewood for it, wait out the days in between. Like Gandalf, look to the achievement popping at first light on the third morning.