The Cave

Well, the achievement description is pretty clear. You have to play an entire game (with three characters) without dying once. This can be pretty hard. My suggestions:

* Take three easy characters with good abilities and a quick and easy puzzle. I recommend: the knight (easy and fairly quick puzzle; special ability lets him be immune to falling damage), the monk (easy puzzle without much opportunity of dying), the scientist (easy puzzle, not much to die for here either), the Hillbilly (can dive under water as long as needed but a long puzzle).
I don't recommend: the time traveller (puzzle too complicated), the twins (puzzle too long), the adventurer (too much possibilites where you can die)

* Most important: Don't fall to your death! If you try rushing too quickly through your adventure, you will fall and die. That's why I recommend going with the knight always head on - you can prevail dying by using your special ability .

* Try avoiding the following, avoidable deaths: staying under water for too long, getting squished by an elevator (happens often enough...), getting squished by the boat in the last puzzle on the island (if you dive underneath the boat, you will die), staying too close to an exploding dynamite, getting squished by a mine cart arriving (happens often enough...)

* If you do die - head to the dashboard immediately! Press the Xbox Guide button , then and quit the game! If you choose to leave ingame, the game will save and you will have to start over again! If you play on Xbox One or Xbox Series, just hit the Guide button and close the game ("menu" button on the tile), this way the game won't save your progress (thanks to Beatmosh for the tip).

* You can skip a complete passage while replaying! In the very first puzzle (where you have to lure the monster to the hot dog on the stick to pull it up with that lever), take a second hot dog with you! Carry it on at all times through the following puzzles (first character puzzle, then the mine carts puzzle, then second character puzzle)! Always carry it on until you get to the zoo. Stop now! Take a character who DOESN'T have the hot dog. Walk with him and he gets "eaten" by the monster. Now, remember the puzzle? Still got that hot dog? You can skip all that recorder getting and so on, just take your character with that hot dog, lure the monster onto the bridge and BAM - you just skillfully skipped an entire puzzle and made your journey a little shorter. ;)

* You can skip two passages (including the above mentioned one) while playing with the monk and the knight which cuts down a lot of time. Thanks to Stavia and SFDATX for this tip: "You can use the monk's telekinesis power to skip the elevator part of the knight's trial by stealing the dragon key through the wall, and to skip all of the hunter (zoo) puzzle by stealing the hot dog out of the machine."

* Be patient. Sometimes it is better not to rush too much in order to avoiding easy deaths. And good luck!

I don't recommend the Knight, his puzzle is a meatgrinder. In fact, the puzzle consist exclusively in exposing your characters to risk. Also, in his puzzle/level, there are several jumps and traps that, while looking easy, are an unnecessary risk due to bad controls.
Go for Time Traveller, Twins and Monk, as their levels are very safe. If you prefer the Scientist, beware the laser.

If an accident happens, use the trick of dashboarding and closing the game (NEVER "Save and Quit" if a character dies). Of course, you can Save and Quit to take a rest if everything is OK.

I did this with the second run recommended team: Hillbilly, Scientist and Twins and I can confirm this works in case you die: pause the game, then Xbox Button and Dashboard out when your character die and becomes smoke.

I even long paused the game before dashboard and when climbed the final stairs with the first character, since the game didn't save, achievement unlocked flawlessly.

You can follow the advices and tips from the Walkthrough here to feel it easier.