The quickest way to rack up answers is to choose the family mode from the main menu. This allows you to play all of the characters in the game against one another using the same controller.
I would suggest picking 10 as the number of characters, set the first one’s difficulty to easy and then spam A for the remaining characters as their difficulty will default to this. Everyone then gets their own question at each level, with the game going on until only 1 person is left. This means that you will get 50 relatively easy questions to reach 1k for each character. Obviously spam A in between answering to speed the process up.
The tracker doesn’t work for this achievement but this game mode helpfully gives you a tally of right answers for each character at the end of the game enabling you to keep track of your progress should you wish to do so.