Thanks for the shout-out ummmm, although in fact I've never been on the Chase. Applied twice but either missed the deadline for the latest round of applying, or was just too far towards the back of the queue.
It's a popular show, anyway, so a lot of people apply, making it perhaps tricky to get through the first stage of submitting an online application (as in Murdo's link above). After that, I assume it works similarly to other quiz shows that I have been on, ie a small quiz + some sort of informal interview. The show is looking for two things: good quizzers, but also people who are "interesting", in one way or another.