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 with Gen 2 on Red 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: Sandshrew, Sandslash, Vulpix, Ninetales, Meowth, Persian, Bellsprout, Weepinbell, and Victreebel.
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 Red 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 number of Gen 1 save files needed two.