Unlike most of the other Reapers, not much is known about Kurenai, as he does not provide any info on his background after he's defeated. All he reveals is that he was once a yakuza hitman, but he was expelled after he lost an important target he'd been following.
After his expulsion, he was offered membership in the Reapers and he accepted to make quick money.
After he joins forces with Ibuki and the Honest Living Association, Kiryu is sent to find Kurenai and convince him to leave the Reapers.
Kiryu finds Kurenai in Theater Square, where he was openly boasting about being a high-ranking member of the Reapers. Once Kiryu approached him, Kurenai offered to show Kiryu "something sweet" nearby. Despite knowing that this was certainly a trap, Kiryu accompanied Kurenai nonetheless. Just as expected, Kurenai led Kiryu to a trap in Children's Park, where several gun-toting henchmen were awaiting. Kiryu proceeded to fight Kurenai and his goons, defeating them all.
After beating him, Kiryu started questioning Kurenai, but the hitman revealed that he wasn't actually a high-ranking Reaper at all: He was one of the newest, least-important members of the organization and he'd set up this trap hoping to capture Kiryu and take him to his bosses so that he'd get promoted. After realizing how much of a failure he is as a hitman, Kurenai agrees to accompany Kiryu to join the HLA.
As a reward for getting Kurenai to defect, Kiryu is paid 100,000 yen.