There are 151 Pokémon in Japanese Blue's Pokédex. Only 150 are needed to get the certificate from the Game Freak employee in Celadon City. The game is programmed to accept any 150, meaning if you use the Mew glitch to get Mew, you can skip another Pokémon. This page focuses on getting the 150 Pokémon that aren't Mew.
In order to complete the Pokédex in Pokémon Blue with only Gen 1 games, you will need access to at minimum a copy of Pokémon Red and a copy of Pokémon Green.
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.
If you have access to Pokémon Stadium, you can get all of these by defeating the Elite Four and Rival.
Otherwise, you will need access to two games (Red and Green) anyway to finish the Pokédex. So, you can have each game choose a different starter and evolve Eevee into a different Pokémon, and then
trade around to get all the Pokémon registered. Make sure that you trade each starter and each of Omanyte and Kabuto at each evolutionary stage!
If you have access to Yellow, you can also just trade all the starters from Yellow.
Four Pokémon evolve by trading: Machoke, Graveler, Kadabra, and Haunter. You can get Gengar and Golem via in-game trade, meaning you will only need to trade for Machamp and Alakazam. The easiest way to get data for them is to trade them to another game and immediately trade the newly-evolved Pokémon back.