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
-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