To gain a Boost chain you need to earn enough Burnout points from pulling off tasks to refil the Burnout Boost Bar whilst boosting, tasks like:
-Driving in oncoming traffic
-Near miss a car
-Jumps
-Barrel rolls
-Flat spins
-Drifts
-Driving through Billboards
-Successfully completing a Stunt Jump
An easy way id suggest to do this is to firstly get in the
GT Nighthawk and head to the Airfield (Details here -
Visit the Airfield in Burnout Paradise) and drift in a circle whilst boosting (
) you can get upto a x3 Boost Chain before the game cancels it.
Alternatively you could head onto I-88 and drive in oncoming traffic whilst boosting.
Here's a video to show you in the airfield