SIDENOTE: Most writers seem to agree that writing the backstory for yourself is important, and this can be an important first step toward the goal you've expressed here. I encourage you to write those backstory scenes for yourself. You say that your characters are all involved in an event, but separately from each other, and that it is backstory--happened prior to the 'real story.' Go ahead and write those scenes.
OK. First, how can you convey the back story? I suggest 'in pieces' throughout your story. Add a detail here and a detail there.
For the sake of this discussion, let's imagine a story of three public figures, and each one is arguing a different side of the gun debate in the US. Imagine the back story is that all three were students at the same university fifteen years earlier, when a mass shooting happened, but each of them had a different experience of that mass shooting, and let's say we don't want that back story to get in the way of the current 'real story.'
In this case, you have three viewpoints and three experiences of a single event. In order to bring the details from that past event into the present-day narrative, you can use a number of tools:
- Perhaps one was wounded by a bullet and walks with a limp. The reason for the limp, with some key dialog or narrative, provides some back story.
- Another character was emotionally traumatized. He is still in therapy--the therapy sessions are a great way to expose part of the back story.
- Another character has a child in school and is terrified about a school shooter--it comes out why.
- A diary is found with details.
- A newspaper runs a 'fifteen years ago on this date' story.
- Another school shooting happens, flooding memories into one of the characters.
But. To ANSWER your stated question of "What care should you take?":
- Keep in mind that readers may not need very much back story. If your story is told well, readers only need a smidge of backstory.
- Keep in mind that using a variety of techniques will help, as above.
- Keep in mind that you will have redrafts and beta readers in your future, and that finishing a draft is a hurdle in its own right, so don't get too bothered if all the pieces don't fall into place straight away.
- Take care to keep the details and timelines consistent between characters.
- Take care not to overuse a single device.
- Take care not to info dump.
There will be other things that you can 'take care to include or avoid' but I hope this provides some guidance as you work through it.