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 Red 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 Sandshrew, two Vulpix, two Meowth, and three Bellsprout 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 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 two.