Though not specifically mentioned, as one of the Hairy Hooligan Viking novice's hunting dragon, Brightclaw was most likely captured from the Dragon Nursery during Dragon Catching Day.
Later, Brightclaw is mentioned as a participant in the Final Initiation Test on Thor'sday Thursday. It partakes of the uncontrollable all-out dragon brawl that breaks out during the test, causing all the Hooligan and Meathead initiates to be exiled for being unable to control their dragons.
After the arrival of the Green Death on Berk, the Hunting Dragons all hide away and decide what to do - Obey or Desert. The dragons are only loyal to their masters when it suits them and "are not the sort of creatures to back a loser." Brightclaw, along with the other hunting dragons, decide to go along with their masters to see who won - the Green Death or the humans. Brightclaw and the other hunting dragons toss feather bombs at the Green Death and Purple Death, while hurling insults at the colossal dragons - all part of Hiccup's plan. Ultimately the two huge dragons kill each other.
Brightclaw is mentioned in name only; no description of its gender or species is given. We do know that it is a Hunting dragon, so most likely the size of a medium to large Dog.
There are a few Hunting dragon species mentioned but not seen on the How to Train Your Dragon Book Website - the Eaglefly and the Dimbruiser. It is possible Brightclaw is one of these species, as they are also not described, but are known hunting dragon species. A Slitherwing is also possible.