A "living dex" is a fan designation for when a player completes the Pokédex and has one of each species in the PC (in Arceus' case, the Pastures) 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 Legends: Arceus is described here. This page will focus more on requirements specific to a living dex.
The difficulty of completing a living dex is dependent on if you choose to have multiple different forms of the same Pokémon at the same time. For instance, the player can get an unlimited amount of Kantonian Vulpix and Ninetales, but since the player only receives a single Alolan Vulpix, trading is needed to have both Alolan Vulpix and Alolan Ninetales at the same time. If form differences are not considered, a living dex is easy to complete and following the instructions on the linked Pokédex page should be sufficient. This page discusses the different forms that are available and what would be required to have all of them simultaneously. It only concerns form differences that can be seen outside of battle (discounting, for instance, Cherrim's Sunshine Form). The locations of the items that enable form changes can be found here.
If form differences are considered, you will need access to some other games. The easiest way to get all the form differences would be to have a second copy of Legends: Arceus, but the game's compatibility with HOME means that compatible Pokémon can be transferred from any game. Enamorus is exclusive to Legends: Arceus, but a shiny Enamorus can be obtained via HOME as a reward for completing the Hisui Pokédex, meaning a second copy or save file of Legends: Arceus is not strictly required.
Only one Alolan Vulpix can be obtained within Legends: Arceus. If you wish to have both Alolan Vulpix and Alolan Ninetales simultaneously, a second Alola Vulpix or Ninetales will need to be transferred from Pokémon HOME or traded from another copy of Legends: Arceus.
The player can obtain an unlimited number of Rotom, and can choose to have multiple Rotom inhabit appliances at the same time. Therefore, with six Rotom, all six Rotom forms (the five appliances and the normal form) can be obtained simultaneously.
Dialga, Palkia, Giratina, Shaymin, Tornadus, Thundurus, Landorus, and Enamorus all have alternate Formes that can be obtained by using an item on the Pokémon. As only one of each Legendary or Mythical can be obtained without trading, if you wish to have both Formes simultaneously, a second species will need to be transferred from Pokémon HOME or traded from another copy of Legends: Arceus.
Technically, Arceus has 19 forms depending on which Plate it is given (18 for the 18 types and the Legend Plate). If you wanted to have all forms simultaneously, you would need access to at least 18 Arceus in addition to the one received in Legends: Arceus itself.
All 28 Unown letters can be found in the overworld in specific locations. After catching all 28, Unown also begin appearing in Solaceon Ruins, and all 28 letters are available there as well.