Here is a list of all the deaths I died before the achievement unlocked. Some deaths listed below might be duplicative or unnecessary (the game is not clear on this).
1. Fall from height - any level
2. Fall into black void - any level
3. Crushed by falling rocks - Chapter 1: First Steps
4. Licked by lizard - Chapter 1: The Great Escape
5. Death by coconut thrown by monkey - Chapter 2: Return!
6. Gored by horny hamster - Chapter 2: Horny Pet
7. Burned to death - Chapter 2: Pyromania
8. Ripped apart by bees - Epilog: The Chase (available earlier?)
Remember, you must die each of these deaths without quitting the application.