The day Midoriya discovered his Quirk was also the day he decided he was meant to keep it secret. Not because of the nature of the Quirk, or the consequences it brought with it, but more because of the excitement and exhilaration it came paired with.
He remembers the day very clearly. It was the day he went to the hospital. The day the doctor brought him the heart shattering news. The boy, the one who wanted to become, more than anyone, a hero, was Quirkless. Later at home he watched the video of All Might again, hoping to gain some hope and inspiration from it, a lifeline, something that would tell him that not everything was lost.
He watched it, heard All Might's famous words, as he started tearing up. In the doorway stood his mother. A look of hurt in her eyes, but not comprehending how much of poor little Izuku's heart would be damaged forever.
'I can't become a hero, can I?' he asked her as the tears were streaming down his face.
She ran over to him and hugged him tight as she said 'I'm sorry' over and over again.
It weren't the words he wanted to hear. Not the ones he needed to hear, and yet she said them. They hurt him, just as badly as the doctor's words did, maybe even more, because they came from the one person he thought would believe in him the most.
He wanted her to hurt just as much as he did, understand the aching pain in his heart, the agonizing pounding in his head, the intense strain put on his entire body from the rush of emotions overwhelming him, when she started to cry out harder. It was almost as if she… finally understood, what he was going through. That she was feeling exactly what he was feeling.
She hurt, just as much as he did, and he enjoyed it. Her loud cries made him feel more wholesome again, her hands clutching her chest, trying to get to the source of the heartache, made him feel… good. At least, better. He didn't understand at first. The doctor told him he was Quirkless. He lacked the extra bone in his foot that separated the Quirkless from the Quirkful.
Midoriya had always been a clever child and knew that something was not quite right. He spent the rest of the week trying to figure out what it was, only coming to the conclusion that it must be his Quirk.
He didn't believe it. First his dreams got shattered, then, right on the same day, he discovered his Quirk. One that would eventually divert him from his original path of trying to become a hero. Who, with such a wicked Quirk, could ever become a hero, anyways? Especially if he enjoyed doing it. After that, as any child would do, he was testing the limits of his Quirk.
At first he thought his Quirk was to bring hurt onto others, but when one day at kindergarten the excitement and joy of the group grew even louder, knowing it was his doing, he changed it to Emotional Enhancer. As he grew older he added something to his notebooks: himself. He wrote down his own Quirk, analyzed it, took it apart bit by bit, to try to understand it to the best extent he could manage. That's when he decided to start experimenting on others.
It's highly frowned upon – read as: illegal – to test your Quirks on others, but seeing as his wasn't even registered, and his Quirk barely noticeable, he thought it would be okay. He tried pushing his limits even further, not only enhancing emotions but also shifting them around completely. Make someone that's happy sad, make someone that's sad happy, discovering to how many people he could limit his Quirk and to what range, eventually making a full scale analysis of his Quirk at age ten, eventually dubbing it "Moodshift".
He thinks back about his childhood friends. Kacchan was the one that hung around him the most, showing off his flashy Quirk while talking Midoriya down about the fact he didn't have any. Before he knew the full abilities of his Quirk, he only used it to enhance, and thus enhancing Kacchan's egotistical behavior, as well as his dominance over Midoriya.
Whenever Midoriya tinkered with Kacchan's emotions, Kacchan became more violent, nearly to the point Kacchan couldn't even understand why anymore. Midoriya loved it when he did that. The burn wounds weren't that welcoming at first, but he got used to it, and eventually started to enjoy that as well, only encouraging Midoriya to do it more often and driving Kacchan into the same hateful behavior, even when Midoriya wasn't doing anything.
Despite all of that, he still wanted to try to become a hero. There were more than one heroes running around that had sadistic traits just like him that did fine rescuing people. He took Midnight as his prime example of what he could be, satisfying both his need to help and his need to hurt.
When it finally became the day of the entrance exam he was nervous like any other person would be. Nobody knew what the entrance exam would be like, so Midoriya was completely in the dark about whether or not his Quirk would prove useful. When thinking about this possibility, still keeping up that positivity, he tripped over his own feet, expecting to plant facedown onto the ground, when a girl quickly tapped his arm and he floated above the ground. He was grateful for the girl but lost his focus.
Once all the hoped-to-become students filed into the auditorium the explanation of the exam began, of course crushing all of Midoriya's dreams once again. The exam was an arena based battle against robots, things without emotions, that couldn't be manipulated by his Quirk.
He doesn't want to think about what happened in there, it's too humiliating. Needless to say, he failed. Zero points. His Quirk wasn't suited for this. His grades in the written exam were perfect though, so they offered him a spot in the General Course. He refused. His heart was too damaged, too many times crushed into pieces and taped back together, that he couldn't stand it anymore.
He told his mother he was going to a school in the heart of Tokyo, and that he was getting a dormitory there, just so he had free range from then on. After he left, he disappeared completely from the face of the Earth for a while. Right up until now, until Kacchan found him. And the rest is history.