He trained to raise the temperature of his flames to meet Endeavor's expectations, but ended up burning his own body. Even when Endeavor stopped him, he ignored the training restrictions and kept going.
To put a stop to it, Endeavor created a new child—that was Shoto Todoroki, born with both fire and ice. After that, he was completely ignored.
One day, while crying alone at his usual training spot, his quirk went out of control, and he suffered severe burns.
He was taken in by an orphanage and slept for about three years.
One day, he received an offer from All For One but refused it. When he woke up and returned home, he found that he had been declared dead. That was equivalent to Toya Todoroki dying and being reborn as Dabi. Seeing Endeavor still obsessed with Shoto, he decided on revenge. Then he joined the League of Villains. Additionally, he sees himself as a failure and Shoto as the success.