Bossk is a Trandoshan, a three-toed, three-fingered reptilian. Like the rest of his species, he has the ability to regenerate lost limbs and can see in the dark. His home planet in the same system as Kashyyyk, the Wookiee homeworld, and like most Trandoshans, Bossk hates Wookiees. In fact, it was a Trandoshan dignitary who convinced the Empire to use Wookiees for slave labor.
The Trandoshans also offered to hunt down any Wookiees that escaped from the Imperials. When his liege lord suggested that he become a bounty hunter who specialized in Wookiee captures and kills, Bossk readily agreed to do so. He outfitted himself with body armor and a variety of weapons, including his favorite weapon, his blaster rifle.
Bossk gained great fame as a bounty hunter, but his most wanted target persistently refused to show himself. The Wookiee Chewbacca--allied with the smuggler Han Solo--had earned a considerable amount of respect in the underworld, and Bossk decided to bring him down. Meanwhile he continued collecting bounties on Wookiees as well as other sentient species.
On one occasion, Chewbacca successfully evaded and Bossk and humiliated him. Several years later Bossk got another chance at the Wookiee and his Corellian friend. Shortly before the Battle of Hoth, Boba Fett gathered several bounty hunters, including Bossk and Dengar, to find Han Solo. They captured him, along with Luke Skywalker and Chewbacca, but with the help of Luke's Jedi abilities, the trio managed to escape.
When Darth Vader put out a call for bounty hunters to find the Millennium Falcon, Bossk went immediately to the Super Star Destroyer Executor. There he found he had some competition from five other hunters: Boba Fett, IG-88, Dengar, Zuckuss, and 4-LOM. Thinking he had an inside track on where the Falcon was located, Bossk took off with some temporary partners who hadn't made Vader's cut. Much later, after being betrayed by his partners, Bossk learned that Boba Fett had succeeded in tracking down the Falcon and delivering Han Solo to Jabba the Hutt.