Start off with well-known DBZ characters such as Piccolo, Vegeta and Goku, three of the most popular characters from the anime series. One way to use these characters would be in training scenes as they prepare themselves for any possible battles that could arise. However, these characters could be used in any number of situations.
Introduce new characters after the DBZ characters are shown. These characters can be Saiyans from distant planets or godlike humans out for revenge or anything else that seems sensible. DBZ characters are so diverse that any kind of character and backstory could fit into DBZ fanfiction.
Have the characters either meet the DBZ heroes in support of their beliefs or challenge them in battle. Make sure the characters feel real. For instance, make the characters have some quirks or flaws. Goku is a lovable goof and Vegeta is a fierce warrior who cannot stand that Goku is the strongest fighter in the universe.
Introduce an already-established villain if the new character is a hero. Maybe Frieza escaped death by morphing into a new form. Maybe Cell regenerated from one blood cell and is more powerful as a result. Or it could be a new villain. Whoever the villain is, just make him or her act like the DBZ characters are at risk.
If the new character is a villain, have him get down to business by attacking the DBZ heroes and the surrounding cities. This allows the new character to feel more like a real threat.
Include multiple fast, intense battles between the heroes and the villains. Likely, most of the heroes will fall to the villain. Only Goku, Vegeta, Gohan and even a new character, the strongest characters, stand a chance against the harshest villains in the series. These characters should not die quickly because if they do, many will not believe it.
A large final battle settles every Dragonball Z saga. Dragonball Z sagas could not end any other way. Fighting is what the anime is all about. Having a final battle in a DBZ fanfiction ending would be a good finality.
Have a celebratory ending. End with a jovial party or an after battle ceremony.