The chain system works like this:
Kill three enemies of one color and then you can kill three more enemies of one color. It doesn't matter the color, they just have to be three of a kind. You can kill three white ships and three black ships for a valid chain. You can kill three black ships and three white ships for a valid chain. You can kill three white ships and three more white ships. Each of these examples are two chains.
The levels always send enemies in patterns of threes, you just need to aim at the right ones to continue the chain.
GS.1000: Halo 3, CoD4
GS.200: BBR, A.Wing, P.Racer