There are 151 Pokémon in Japanese Green/International 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 Green/Blue with only Gen 1 games, you will need access to at minimum a copy of Pokémon Red.
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 an unlimited number of these by defeating the Elite Four and Rival repeatedly.
Otherwise, you will need multiple files. For Hitmonlee/Hitmonchan and Omanyte/Kabuto, you will need access to Red anyway to complete the Pokédex, so you can pick one of them in Green/Blue and the other in Red.
However, for the starters and Eevee's evolutions, the trade process is a little more complicated.
If you have access to two other games, then each game can pick a different starter and evolve Eevee into a different evolution, and you can trade around to get them all registered in your Pokédex.
If you have access to Yellow Version, you can also just trade all the starters from Yellow.
If you only have access to two games total (Green/Blue and Red), you would need to do the following:
Now you will have two starters and two of Eevee's evolutions on your first file and the third starter and Eevee evolution on the second. At this point, trade back and forth
until the game you're trying to complete the Pokédex on has data for each starter and all of Eevee's evolutions.
Note that you will need to trade your starter, Omanyte, and Kabuto at each evolutionary stage, because you only have access to one of each species.
This is why I recommended keeping your starter unevolved in the steps above, especially if you're also trying to complete the Pokédex in the second game.
Wait until you have all three unevolved starters available, trade them, and then you can go ahead and evolve them (and trade again).
If you have access to Gold, Silver, or Crystal, you wouldn't need to worry about this due to breeding. See the Green/Blue and Gen 2 page for more information.
Four Pokémon evolve by trading: Machoke, Graveler, Kadabra, and Haunter. The easiest way to get data for them is to trade them to another game and immediately trade the newly-evolved Pokémon back.