He was betrothed to Catelyn Tully, even though he apparently didn't want to marry her originally. At the time of the betrothal, he was already in a relationship with Barbrey Ryswell, whose maidenhead he took one night. Both she and Eddard Stark remember Brandon as a fiery young man who had no problem taking what he wanted.
However, when the marriage was announced, Petyr Baelish spoke out against it, declaring his undying love for her. He challenged Brandon to a duel, and Brandon accepted; when Petyr showed up in scant armour, Brandon promised Catelyn that he would not kill the boy, and even removed some of his armour. Throughout the fight, he offered several chances for Petyr to back down, which fell on deaf ears. However, the duel was a savage event, and Brandon ended it when he left Petyr with a wound so vicious that it almost killed him.
During Robert's Rebellion, he was executed in King's Landing alongside his father by the Mad King when he entered the Red Keep with a handful of companions (Jeffory Mallister, Kyle Royce, Ethan Glover, and Elbert Arryn) demanding that Crown Prince Rhaegar Targaryen answer for his supposed abduction of Lyanna. The five young men were arrested by King Aerys II and charged with treason. Their fathers were called to King's Landing to answer these charges as well but were apprehended on arrival and presented in front of the court in the Red Keep.
Lord Rickard was tied to a stake and Brandon to a strangulation device. Brandon strangled himself reaching for a sword inches away from him in an attempt to rescue his father, who was burned alive by wildfire in front of him, on orders of the Mad King, who died first is unclear.
After Brandon's death, Catelyn married Eddard instead. Brandon was buried with the Kings of Winter in the crypt beneath Winterfell. Jaime Lannister remembers the monstrous nature of his death even years later, and recalls it to Catelyn in A Clash of Kings after establishing that he killed King Aerys, and she was once betrothed to the young northerner.