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 mostly-completed Pokédex on HeartGold is described here. This page will focus more on requirements specific to a living dex using only HeartGold and SoulSilver.
You will need access to at minimum three separate HeartGold and SoulSilver save files in order to obtain all starters. Each file will need to progess as far as defeating Red in order to access the Hoenn starters. Since you can breed them, only three files are required.
This is just to note that if you are attempting to complete a living dex on both HeartGold and SoulSilver simultaneously, you will need at minimum four save files, in order to ensure that each copy of HeartGold and SoulSilver gets access to Latios, Latias, Groudon, and Kyogre.