He begins his tale by stating that Tancred was the brightest offspring of a shining lineage and was even more sublime than his maternal uncles. (including Bohemond) The whole Guiscard family was just awesome and glorious and they kicked the Byzantine emperor's butt but Raoul was not going to tarry about on this awesomeness because he had to get into his topic: The princely genius of Tancred disposed itself to holy war. (and that was awesome, too.)
Tancred wasn't like all the other little boys. He was serious, man. He was zealous in his studies, insatiable to be taught, remembered everything he was told, more agile than the other youths, more serious in mind than any old man, able to leap over tall buildings with a single bound! He didn't spare his own blood; he certainly was not going to spare any of the enemies'. He signed up right away when the call went out from Pope Urban for a crusade. Before or after Bohemond, enquiring minds want to know.
He did seem content to live in his uncle's shadow; he does not appear to have coveted a kingdom for himself. Bohemond appointed Tancred regent over Antioch when he was away and Tancred never tried to usurp him. He was noted for having given his banner to some Arabs who had taken refuge on the roof of the Temple of the Mount, which placed them under his protection. They were killed anyway and Tancred was reported to have been very angry about this. I wonder if it was a matter like when the crusaders took Marra and Bohemond told some Saracens that if they wanted to live, they should gather up their belongings and family and go to a palace near the gate. (Gesta Francorum) After the fighting was over, Bohemond took all their gold and precious items, killed some and sold the others into slavery. Maybe Tancred was angry that this potentially lucrative slave money was spoiled for him rather than because he had any pity for the people.
An Arab, writing during the Third Crusade, Usamah ibn Munqidh, tells a story that during a truce between Shaizar and Antioch, Tancred asked for the gift of a horse. The horse was delivered by a young Kurd named Hasanun, who rode that horse to victory in a race. Tancred bestowed a cloak of honour on him and he, in return, requested that if he should be taken prisoner by Tancred that his life might be spared. Later he was captured and Tancred ordered his right eye to be put out and his life ransomed. Glorious. Not one of his finer moments but then, in an article, T.S Asbridge tells how Usamah also related a story about Robert fitz-Fulk the Leper who was close friends with Tughtegin, the atabeg of Damascus and this friendship did not prevent Tughtegin from personally beheading Robert when he was captured in 1119. Walter the Chancellor added that Tughtegin had a gold and jewelled cup made out of Robert's skull.