The hunting dogs were chasing the bear for all they were worth.

Literal

Hunting-dog [topic-は] bear [object-を] desperately chasing-ていた.

猟犬 ('hunting dog, hound'). 必死 ('desperate, frantic') + に (adverbial) = 'desperately.' 追いかける ('to chase, pursue') is a compound verb. ~ていた for past continuous ('was chasing').