Conarium

I strongly recommend following the walkthrough for full details on where and how to get each death. In brief, they are:
1 - Crash into a wall 4 times while piloting the submarine during the "Underwater Caves" segment
2 - Anytime you encounter a passage blocked by ivy, chop at it with your axe until the poisonous gas kills you
3 - Anytime from the "The Entrance" segment and beyond, after you've acquired the ornamental object that you can carry to get past ivy, enter a tangle of ivy while holding the ornamental object then just stand in place and wait ~60 seconds for the object's blue charge to run out. The ivy will grow back around you and kill you.
4 - Towards the end of "The Entrance," after you've drawn the glyph and taken the rustic elevator down to the other level, during the section where you have to run away from the mummies just wait and let a mummy catch you.
5 - In the "Desert Ruins" segment, after you've completed the star map puzzle and are ready to the leave the area, there will be a stronger earthquake and the character will comment that you need to hurry and get out. Start to head out until you trigger the second strong quake and mini-cutscene, then just stand still and let the giant blob catch you.

The most important thing to note is that the achievement will NOT pop right after you get the fifth death. You must finish the game after this point, and only once the credits start rolling will you get the achievement. Although it doesn't count towards the five deaths, I recommend drinking the substance next to Dr. Faust to get the short ending.

I have not tested whether all five deaths need to be in the same playthrough, or if you can unlock this across a couple playthroughs, but I would recommend keeping it all on the same playthrough to be safe. It takes less than an hour to speedrun through an all-death run if you have a previous save from the underwater caves segment that you can load up and start from.