To unlock the grayed-out characters, complete a game of Tamagotchi Party On!. For example, if you choose a 3-election game, you will not get a character for each election you complete, however you will for completing the entire 3-election game itself. If you want to unlock characters faster, just keep playing 1-election games.

To unlock the final 3 hidden characters, select the 'Challenge' option on the main menu.

Do not press A to skip the CPU's roll. If you do, instead of relying on the die'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.