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 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 nine separate Gen 1 save files.
If you have access to Yellow, this number changes to four separate save files: one Red file and three Yellow files. With Yellow, you can collect each starter on each of the three files and trade it to Red, as well as Eevee's three evolutions and three of Omanyte, Omastar, Kabuto, and Kabutops, with the Red 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: