Scoring 5,000+ points in a daily challenge is based on the number of correct answers and the amount of time it takes for you to give it. You have 6 questions and you'll get to choose the categories at the start of the challenge. I got this after getting 5/6 answers right in a quick fashion. It helps if you go for the other achievements first to learn which categories you do best in and you will also start to memorize the answers.
You can do this once a day so just make sure each day you go for other achievements, you go for this one. If you are getting the points but the achievement isn't popping, try doing a hard reset on your xbox. I got over 5k a couple times before I did a hard reset, but it popped the very next time after the reset.