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Day 1: skip the tutorial, run to the camp collecting all the coins, build a camp. Wait for the night to come. Leave one coin on both sides of your camp.
Day 2: collect the coins from the chest, run to either side until you get to the treasure chest and get the coins. Return to the camp. Hire a builder, and build a wooden barricade on both sides of your camp. At night leave one coin on both sides of your camp.
Day 3: collect the coins from the chest, run to the side you didn't run to on Day 2 until you get to the treasure chest. Return to the camp. At night leave two coins on both sides of your camp.
Day 4: collect the coins from the chest. Wait until night. Leave three coins on both sides of your camp.
Day 5: collect the coins from the chest. It's time to prepare for the Blood Moon! All the monsters will come from one side at night. Wait until you see them coming and run in the opposite direction leaving all your coins behind to distract the pursuers. You should have enough coins to let you escape safely.
When the Day 6 dawns this achievement will unlock.

Note: The part that involves looking for the treasure chests, hiring a builder and building the barricades is completely optional. You should be able to escape using just the coins from your camp. But doing it my way guarantees that the monsters won't be able to reach you on Day 5.