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 Yellow 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 the breeding mechanic in Gen 1. You will need all of the following: Weedle, Kakuna, Beedrill, Ekans, Arbok, Meowth, Persian, Koffing, and Weezing.
Also, if you are playing an International version of Yellow, you will have to trade your only Jynx from either Red or Blue. Therefore, if you are trying to simultaneously complete a living dex in Yellow and another Gen 1 game, be sure to "save" at least one Jynx so that both files can have a Jynx.
You will also need to trade for a second Pikachu or Raichu, so that you can permanently keep a Raichu.
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 Yellow alone, you will need at minimum four separate Gen 1 save files, at least three of which would need to be on Yellow. Your trade process should look something like this: