A "living dex" is a fan designation for when a player completes the Pokédex and has one of each species in the PC or the party at the same time. It's often considered to be much more difficult and time-consuming than a regular Pokédex because it often involves, for instance, playing the same game multiple times to get multiple copies of a Legendary Pokémon.
The method by which you can get a completed Pokédex on Japanese Green/International Blue is described here. This page will focus more on requirements specific to a living dex.
You will need to trade for multiple members of a version-exclusive species' evolutionary family, instead of just one. This is due to the lack of a breeding mechanic in Gen 1. You will need all of the following: Ekans, Arbok, Oddish, Gloom, Vileplume, Mankey, Primeape, Growlithe, and Arcanine.
To have a living dex, you will need all of the following on your file:
If you have access to Pokémon Stadium, you can get an unlimited number of these by defeating the Elite Four and Rival repeatedly.
Otherwise, to complete this living dex on Green/Blue alone, you will need at minimum nine separate Gen 1 save files.
If you have access to Yellow, this number changes to four separate save files: one Green/Blue file and three Yellow files. With Yellow, you can collect each starter on each of the three files and trade it to Green/Blue, as well as Eevee's three evolutions and three of Omanyte, Omastar, Kabuto, and Kabutops, with the Green/Blue file providing Eevee and the last Fossil Pokémon. In this scenario, each of the starters traded from Yellow can be evolved to a different form.
Without access to Yellow, your save files would have to look something like the following: