Neon White

You CAN get the achievements for both endings on one save file

To gain access to the option to write Green into the Book of Life you'll first need to collect every gift and complete your relationships with the Neon characters to unlock all of White's memories, gifts can be found in almost every level of the game, you can replay a mission to find the gift by visiting the Job Archive from the Heaven's Gate option in the main hub

Edit: ChiflaGoodluck is correct, you need only to complete your relationships with the Neon characters, Mikey and Raz aren't relevant for this achievement

To get this achievement, or the corresponding achievement for writing Green into the Book of Death you need to complete the last chapter by selecting "Start Mission" from the Heaven's Gate area of the hub, do NOT use the Job Archive as this will not grant the achievement

Once you've completed the final level you'll be given an option to write Green into the Book of Life or the Book of Death, simply pick the Book of Life option and the achievement will be granted.

Have fun, Neons!

In order to obtain this, you must acquire all of White's memories, which can be done by acquiring gifts and using them to gain insight into the other Neons. Then you will be given this choice after playing the final mission.

WARNING: Do not choose this option unless you have already chosen the "Book of Death" option.

Unknowingly, I went through the game completing everything 100% as I went through my first go, and therefore was given the choice of either Book of Life or Book of Death when I finally finished. After choosing Book of Life, the game bars you from making a choice again, therefore requiring you to replay the whole campaign in order to get the "Book of Death" achievement.