Tamagotchi: Party On!
To unlock the grayed-out characters, successfully complete a game of Tamagotchi: Party On!. For example, if you choose a 3-election game, you will not unlock a character for each election you complete; however, you will unlock one for completing the entire 3-election game itself. If you want to unlock characters faster, just keep playing 1-election games.
Get Gold medals on all the "Mini-games" and "Gotchi Games" in Challenge mode to unlock the last three secret characters in all modes: Purimatchi, Matsuritchi, and Shitiketchi.
Do not press A to skip the CPU's roll. If you do, instead of relying on the dice's bounces to choose the amount of spaces they move, it will be a random number, and could result in giving the CPU more spaces to move than they would have had before.