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 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.
However, not all 249 Pokémon that aren't Mew or Celebi are available in Gold, Silver, and Crystal without glitches or (defunct) events. In order to get all 249 Pokémon, trading with Red, Green, Blue, or Yellow is required. This page focuses on getting as many Pokémon as possible with only Gold, Silver, and Crystal.
The maximum number of Pokémon you can get this way is 232, or 233 if you are able to access the GS Ball Celebi in Crystal.
In order to complete the Pokédex in Pokémon Silver this way, you will need access to at minimum a copy of Pokémon Gold. Crystal is only required if you are able to access the GS Ball Celebi.
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 Crystal, where you can get infinite amounts, and/or Mystery Gift will allow for more to be obtained.
If you have access to Crystal or Mystery Gift, you will need at minimum three separate save files to complete the Pokédex. Each save file will get a different starter, and they can be bred and traded as needed. Crystal or Mystery Gift can be used for obtaining the stone evolutions.
If you only have access to Gold and Silver and do not have access to Mystery Gift, you will need at minimum two games (Gold and Silver) and four separate save files to complete the Pokédex. You will need to do the following:
Keep in mind that two of the Pokémon that evolve via Fire Stone, Growlithe and Vulpix, are version-exclusive to Gold and Silver, respectively.
If you have access to Red, Green, Blue, and Yellow, you will still need a minimum of three save files in order to get the starters, but the stone problem is a non-issue since infinite amounts can be bought in those games. If you have that access, you can follow the instructions here.
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.
The remaining Pokémon are obtainable only in Red, Green, Blue, or Yellow, or in defunct events.