From GameFaqs:
To get Luigi:
Beat the "Bonus 1 Practice" game found in the single player menu with the
original 8 characters. This includes Yoshi, Mario, Samus, Kirby, Link, DK,
Pikachu, and Fox.
To get Jigglypuff:
Beat the game on ANY difficulty, with ANY number of lives, and with ANY
character. Use as many continues as you like. Beat Jigglypuff at the end of the
game to gain the ability to use her.
To get Captain Falcon:
Beat the game in less than 20 minutes. DO NOT use any continues as this would
disable any chance of fighting Captain Falcon at the end of the game. Beating
him gives you the ability to use him. The bonus games count too, so you should
probably just kill yourself on the first two and save yourself some time. It
doesn't matter if you get him with, or without the timer activated.
To get Ness:
This is challenging. Set the game difficulty to "Normal" and set the number of
lives to 3. Beat the game without any continues and you will have a chance to
fight Ness. Beat him and you can use him.