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.
With Gen 1, the Legendary Pokémon are not a concern, and with Gen 2, neither is breeding.
The method by which you can get a completed Pokédex with Gen 2 on Japanese Green/International Blue is described here. This page will focus more on requirements specific to a living dex.
This is just to note that you will need to trade for multiple version-exclusives, instead of just one. You will need two Ekans, three Oddish, two Mankey, and two Growlithe from your other games.
The real issue with a Gen 1 only living dex lies in the mutually exclusive Pokémon. 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 all of these by defeating the Elite Four and Rival.
Otherwise, to complete this living dex on Green/Blue alone, you will need at minimum three separate Gen 1 save files and one Gen 2 save file. Your process should look something like this:
If you have access to Yellow, the first two steps can be completed in one file on Yellow, making the minimum two.