One way to start a Dragon Ball Z launch fanfiction is to consider the setting. The planet Earth is constantly under threat in Dragon Ball Z. You could start your story with a new, unknown danger approaching Earth. It could be a powerful alien race that no one has ever heard of before. Then, introduce how the Z - fighters, like Krillin, Yamcha, and Tien, respond to this threat at the very beginning. They might be much weaker compared to later in the series, so it adds an interesting dynamic to the story.
Well, first you can start by thinking about the characters you want to focus on. Maybe you're a big Goku fan, so you could center your fanfiction around him. Then, consider the plot. It could be a new adventure, like Goku finding a new form in a different universe. Another way is to look at the relationships between characters. For example, how Vegeta and Goku's rivalry could develop in a new way.
First, think about the family members. Decide who the main characters in the family are, like a Saiyan family perhaps. Then, consider their personalities. Are they strong - willed like Goku or more laid - back? Next, create a plot. It could be about protecting their family from a new threat in the Dragon Ball universe.