Dabi's backstory has yet to be explained in the anime, so if you haven't caught up with the manga, keep scrolling.
Known originally as Toya, Dabi is Enji and Rei Todoroki's eldest son. While he didn't inherit his mother's ice-based abilities, he did inherit her physical frailty and ice resistance. He also inherited a version of his father's fire quirk that did not include immunity to fire. Using his quirk did serious damage to his body, but because his father made it clear that the only way to gain his love and respect was to be strong, he insisted on training even when his parents told him to give up. He continued training for years, becoming increasingly desperate and prone to lashing out. This got worse when Shoto was born, since his father finally had the child he actually wanted.
His father never gave him the actual emotional support he would have needed to process the fact that he couldn't properly use his quirk, and he created the environment that made him so desperate to become a hero in the first place. Toya's continual training left him covered in horrible burns and suffering from constant physical and emotional pain. Eventually, Toya supposedly perished by overheating his body to 2,000 degrees Celsius. He somehow survived and ended up with the League of Villains.