This is a basic guide that will unlock the 4 difficulty achievements and the 4 no death achievements without needing to play through the Nightmare difficulty.

The achievements:

• Junior Investigator - Beat the game on Easy.
  
• Paranormal Investigator - Beat the game on Normal.
  
• Elite Investigator - Beat the game on Hard.
  
• Legendary Investigator - Beat the game on Nightmare
  
• Unscathed - Do not get killed by a ghost on Easy difficulty.
  
• Unshaken - Do not get caught by a ghost on the Normal difficulty.
  
• Unbeatable - Do not get caught by a ghost on the Hard difficulty.
  
• Ghostbuster - Do not get caught by a ghost on the Nightmare difficulty.

I tried to finishing Nightmare properly, but the monster spawns require too much luck. Nightmare also doesn't save so you need to start from scratch every time you die.

The process involves tricking the game by backing up your save and reloading it after a fresh new game.

1. Play through the game (on any difficulty) until the last boss. There's a checkpoint at the start of the battle. It doesn't matter how many times you die in this step.
   
2. Backup your save now. It is located in ...\AppData\LocalLow\Stormlord Games\Curse\
   
3. Finish the game now and get the Ledger/second ending achievements.
   
4. Start a new game on any difficulty (I'm not sure how much of this playthrough you need to play, I played through a fair bit for "Tasha’s Warning"). It's important that you don't die here.
   
5. Quit back to main menu.
   
6. Restore the save from step 2.
   
7. Continue from the main menu. Done correctly, you'll be at the start of the final boss.
   
8. Defeat the boss without dying.
   
9. When it says "Edgar is defeated!", immediately switch to Nightmare from the pause menu.
   
10. You should unlock all the difficulty and no-death achievements after you are back in the chapel and put the dagger in your backpack.

Note: if you don't want to fiddle with save files you can still do Step 9 and 10 to unlock the just difficulty achievements.
Additionally, I'm not sure if playing a no-death run on Easy then switching difficulties at Step 9 will work for the no-death achievements on higher difficulties. Might be worth testing if you aren't comfortable with savescumming.

The game seems to check the difficulty you're playing only at the very end when it awards you the achievement. So you can switch to Nightmare any time before the check and you'll unlock the difficulty achievements.

For the no-death achievements, the game uses Steam stats to log your deaths, and not the local save file. This prevents you from backing up your save before every battle and savescumming in the traditional way. Starting a new game in step 4 resets your deaths to 0, and continuing your previous save at step 7 essentially puts you at the final boss with 0 deaths.

This may not work anymore after the game was updated in May 2018. These don't seem to unlock:

• Unshaken - Do not get caught by a ghost on the Normal difficulty.
  
• Unbeatable - Do not get caught by a ghost on the Hard difficulty.
  
• Ghostbuster - Do not get caught by a ghost on the Nightmare difficulty.

I think some new stats were added too (I don't really remember), so the game may have additional checks for those achievements now.