In many such fanfictions, Draco may try to make amends for his past behavior towards Hermione. He could start by apologizing sincerely for all the unkind things he's said or done. Then, he might try to integrate into Hermione's social circle more. For instance, he could volunteer for the same causes as Hermione, like working on elf rights or something similar. This would give him more time to be around her and show her that he's truly a different person now. He may also give her gifts that are meaningful to her, like a special quill that has some sort of magical property related to her favorite subject.
Well, often in these fanfictions, Draco tries to win Hermione back by using his family's influence in a positive way. For example, he might use his connections to get Hermione access to rare magical books or artifacts for her research. He could also try to bond with her over shared interests. Since they're both very intelligent, he might engage her in deep discussions about magic theory or the history of the wizarding world. And of course, there's the classic love - letter - in - a - bottle - sent - with - a - charm scenario, where he pours out his feelings for her in a very romantic way.
One common scenario could be that Draco realizes his past mistakes and starts to show his changed self to Hermione. He might start by being kind to her in small ways, like helping her with her studies in a subject he's good at. Another could be that he defends her when others are being mean. Maybe he stands up to some pure - blood supremacists who are bad - mouthing Hermione, which starts to change her view of him.