Brilliant Diamond has two separate Pokédexes to complete: the Sinnoh Pokédex and the National Pokédex (which consists of only the first 493 Pokémon in the National Pokédex rather than the total 898 available when the game was released). All Pokémon from the Sinnoh Pokédex, except for Manaphy, are available between Brilliant Diamond and Shining Pearl. Additionally, all Pokémon from the National Pokédex, except for Mew, Celebi, Jirachi, Deoxys, Phione, Manaphy, Darkrai, Shaymin, and Arceus, are available between the two games. Because Brilliant Diamond and Shining Pearl can communicate with Pokémon HOME, there are effectively three separate Pokédexes that the player can complete for Brilliant Diamond:
Any Pokémon captured in Brilliant Diamond and Shining Pearl and transferred to Pokémon HOME (except for Spinda, which cannot be transferred) will additionally count for the HOME National Pokédex.
Brilliant Diamond and Shining Pearl can both transfer Pokémon to and receive Pokémon from Pokémon HOME, so long as the Pokémon is contained within the 493 Pokémon of these games' National Pokédex. They cannot receive Pokémon that have ever been in Legends: Z-A.
Since all 493 Pokémon in these games' National Pokédex can be obtained from other games via Pokémon HOME (including, for instance, a copy of Diamond, Pearl, or Platinum for the original DS, via Pokémon Bank), there are multiple games that can provide Brilliant Diamond with the Pokémon it is missing. However, the easiest way to get the missing Pokémon is simply to use Shining Pearl, which contains all of the missing Pokémon apart from Mythicals, and which can directly trade with Brilliant Diamond to trigger trade evolutions.
The following version-exclusive Pokémon have to be traded from another game (all of them can be found in Shining Pearl):
17 Pokémon evolve by trading: Golem, Machamp, Alakazam, Gengar, Politoed, Slowking, Steelix, Scizor, Kingdra, Porygon2, Huntail, Gorebyss, Rhyperior, Electivire, Magmortar, Porygon-Z, and Dusknoir. The easiest way to get data for them is to trade them to another game and immediately trade the newly-evolved Pokémon back.
You will need two King's Rocks, two Metal Coats, one Dragon Scale, one Up-Grade, one Deep Sea Tooth, one Deep Sea Scale, one Protector, one Electirizer, one Magmarizer, one Dubious Disc, and one Reaper Cloth to evolve Politoed, Slowking, Steelix, Scizor, Kingdra, Porygon2, Huntail, Gorebyss, Rhyperior, Electivire, Magmortar, Porygon-Z, and Dusknoir.