When you write a biography (bio), ask yourself who you are writing for and what you want to communicate. What does your audience care about? What will make the bio relevant to them? A good bio forms a deep connection with the reader so that they will be prepared to hear or read further information from the subject.
Consider what is important to the reader and emphasise those points. If the bio is introducing a scientist, their academic credentials and important discoveries or publications will be relevant. If the bio is introducing a movie star, use a completely different tactic of referencing their famous films or stories about them. In either case, be sure to carefully understand what the audience is looking for.
Keep the material short. The bio is meant as an introduction to a person, not a tome about their entire life. Keep the bio to several sentences or at most a few paragraphs. The audience or reader's attention will be held during that period and you can cut right to the point of the bio.
Refrain from details that your audience will not be interested in. This goes hand in hand with keeping the bio to a short length. If the bio is for an athlete, the audience will probably not want to know about the person's accounting studies after retiring from sports. Of course, the details versus extraneous material is a judgment call which gets better with experience.
Conclude with a personal anecdote. This personalises the bio and makes the audience feel a connection to the person. A simple personal note might be "Mr. Smith is from London, is married with two children." However, a more detailed anecdote can also be included that contains a story with a relevant meaning for the audience.