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 Let's Go Pikachu 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 a breeding mechanic. You will need all of the following: Oddish, Gloom, Vileplume, Mankey, Primeape, Sandshrew, Sandslash, Grimer, and Muk. If you are trying to collect all Alolan forms, you will also need two Alolan Sandshrew and two Alolan Grimer. If you have access to Pokémon GO, you will also need Meltan and Melmetal separately.