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 HeartGold is described here. This page will focus more on requirements specific to a living dex.
You will need access to at minimum three separate Diamond, Pearl, or Platinum save files in order to obtain all Sinnoh starters.
You will also need access to at minimum three separate HeartGold, or SoulSilver save files in order to obtain all Kanto and Johto starters. Each copy of HeartGold or SoulSilver will need to progress as far as defeating Red to obtain the Kanto starters.
You will also need access to at minimum three separate Ruby, Sapphire, Emerald, HeartGold, or SoulSilver save files in order to obtain all Hoenn starters. If HeartGold and/or SoulSilver are used, each copy will need to progress as far as defeating Red to obtain the Hoenn starters.
If you intend to complete the living dex in all five games, you will need the following:
Additionally, an appropriate number of Clamperl must either be evolved in Ruby, Sapphire, and Emerald and sent to Diamond, Pearl, and Platinum, or the player will need to obtain three DeepSeaTooths from wild Carvanha or Sharpedo and three DeepSeaScales from wild Chinchou, Lanturn, and Relicanth.