The Dark Tournament was one of the highlights of Yu Yu Hakusho's run, with the four teens and Genkai being forced to fight for their lives against teams of powerful demons. The catalyst for their situation is a pair of brothers known as the Toguros, the younger of which has a past with Genkai that is slowly revealed throughout the arc. He looms in the background of the tournament, mostly just to intimidate Yusuke before their fateful fight in the finals. His intentions are sinister, to say the least. Or at least, it initially seems as though they are.
First introduced when Yusuke, Kuwabara, and Botan rescued Yukina from the grasp of Tarukane, the Toguro Brothers are initially bodyguards to the greedy human before the younger Toguro, the bigger brother, decapitates Tarukane with a single punch. The pair are later shown to be working for another human named Sakyo, who wants to destroy the barrier between the Human World and Spirit World. They purposefully lost the fight against Yusuke and Kuwabara so that Sakyo could win Tarukane's fortune in a wager.
It's the younger Toguro that "invites" Yusuke and his friends to participate in the Dark Tournament, which they are unable to refuse on the threat of death. Initially, he seems to be motivated by pure entertainment value -- simply looking for a fun fight. However, when taking into account his backstory and the knowledge that Yusuke would ask Genkai to be the fifth member of their team, it might go deeper than this In their younger days, Genkai and Toguro were lovers who trained together. They even participated in the Dark Tournament after a demon named Kairen killed and ate their pupils and colleagues. The Tournament left both Toguros mentally injured, so they decided that their prizes for winning would be to become demons. 50 years later, he would kill Genkai before the Final Round, enraging Yusuke.
This could be seen as Toguro seeing potential in Yusuke, pushing him beyond his current limitations in order to get the best fight out of him. He's considered one of the strongest demons in existence, after all, with few being able to walk away from a brawl with him. He's bored, and he knows that if Genkai is training Yusuke, then he must have enormous potential. However, Yusuke's strength is powered by emotions, which is confirmed by Genkai when she possesses Puu and tells Toguro to kill one of Yusuke's friends in order to bring out Yusuke's true power. It was only after Yusuke broke the Spirit Cuffs Genkai had placed on him, limiting his power, that Toguro was at 100 percent strength.
Toguro "kills" Kuwabara and Yusuke is able to find the strength to launch one final Spirit Blast at Toguro, who also goes beyond the full strength he initially revealed. Rather than dodging the attack, he takes the full brunt of it, resulting in his body crumbling away from pushing it too far. This is where his true motivations for fighting Yusuke becomes a bit murky.
At this point, Toguro's been proven to be able to avoid Yusuke's attacks numerous times throughout their various fights. He's also been able to defeat Genkai, who is possibly the strongest human alive at that time. So why didn't he dodge Yusuke's final attack? It seems that Toguro wanted to die during this fight. It's speculated that he felt shame for how low he had fallen since becoming a demon, and a conversation with Genkai after his death points to this being the case. He also chooses to be punished for 10,000 years for 10,000 cycles in Hell, even after Koenma tells him that he doesn't have to go through that sort of torment.
Though the reasons aren't entirely clear, Younger Toguro became a demon so that he could be the strongest being alive. It could be because he initially wanted to protect those close to him so that they wouldn't end up like his students, but because of the trauma he went through during the Dark Tournament, that goal became distorted. It morphed and changed into just wanting to fight so that he wouldn't be bored with his existence, and then changed again to wanting to die to atone for his sins after becoming a demon.
He wanted to die at Yusuke's hands not only because he not only saw the potential the boy had, but also because Yusuke embodied the ideals that Genkai held. Having Yusuke be the one to defeat him in Yu Yu Hakusho would also prove that Genkai's hope in humanity was well-placed; stronger than Toguro's belief that true physical strength was all that mattered. In the end, he accepts his loss gracefully.