Why is my Bunn overflowing?
If I understand your description, it sounds as if water is flowing into the basket more quickly than it is flowing out. This could require adjusting the nozzle feeding the basket or the dispenser on the basket; probably the former, from your description of how soon after brewing the problem starts. Is there any way to slow down the inflow?

According to the Bunn troubleshooting guide, it is also possible that the coffee is too finely ground, or that the filter is not as permeable as it should be - naturally, Bunn recommends you use their brand of filters, and if you aren't, that's an easy thing to try. They also suggest that the sprayhead may be missing (it could also just be loose), and that you avoid finely ground coffee, decaffeinated coffee, and soft water.