For most commercially produced bagel chips:
Brocho before eating this: Borei Minei Mezonos
Brocho after eating the proper shiur of this: Al Hamichya
If one is Koveah Seduah on bagel chips (and they are not fried), Hamotzi Lechem Min Ha'aretz and Bentching are required.
However, if the bagel chips are made from leftover bagels:
Brocho before eating this: Hamotzi Lechem Min Ha'aretz
Brocho after eating the proper shiur of this: Bentching
Source: Star-K (Kashrus Kurrents Volume 25, No. 4)