There are 251 Pokémon in Silver's Pokédex. Only 249 are needed to get the certificate from the Game Freak employee in Celadon City. The game is programmed to accept any 249, meaning if you use the Mew glitch or Celebi egg glitch to get Mew or Celebi, or trade Celebi from a copy of Crystal that obtained it via the GS Ball event, you can skip up to two other Pokémon. This page focuses on getting the 249 Pokémon that aren't Mew and Celebi.
In order to complete the Pokédex in Pokémon Silver, you will need access to at minimum one of the following combinations:
The following version-exclusive Pokémon have to be traded from another game:
Several Pokémon are mutually exclusive, meaning that if you get one species, you cannot get at least one other species. The list of mutually exclusive species is below.
Pokémon that evolve via Fire Stone, Water Stone, Leaf Stone, or Thunderstone are listed here because only one of each stone can be obtained in Gold and Silver. Access to Red, Green, Blue, Yellow, or Crystal, where you can get infinite amounts, and/or Mystery Gift will allow for more to be obtained.
If you do not have access to Crystal or Mystery Gift, keep in mind that you will need to trade your stone-evolving Pokémon to your copy of Red, Green, Blue, or Yellow in order to evolve them.
Regardless, in order to get all three starters, you will need access to at minimum three separate save files of Gold, Silver, or Crystal, where each save file picks a different starter.
Within Red, Green, Blue, and Yellow, some Pokémon are further mutually exclusive:
If you have access to Pokémon Stadium, beating the Elite Four and Rival will reward you with a random Pokémon that can include the starters or Fossil Pokémon. These Pokémon can be traded first to a copy of Red, Green, Blue, or Yellow and then to Gold, Silver, or Crystal.
If you cannot access Pokémon Stadium, in order to get both Fossils, you will need access to at minimum two separate save files of Red, Green, Blue, or Yellow, where each file picks a different Fossil. If you do not have access to Yellow, you will need three files instead.
Ten Pokémon evolve by trading: Golem, Machamp, Alakazam, Gengar, Politoed, Slowking, Steelix, Scizor, Kingdra, and Porygon2. 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, and one Up-Grade to evolve Politoed, Slowking, Steelix, Scizor, Kingdra, and Porygon2. Note that these cannot be obtained by trading with Red, Green, Blue, or Yellow.