In my post-apocalyptic story, my narrator keeps a journal, and each new chapter starts with a journal entry of hers that's relevant to the events of the chapter. How often is this done, and is this a good idea to do? I'm actually overhauling the story and reconsidering whether or not I want to continue to write her little entries.
These are called epigraphs, and they're perfectly fine
Off the top of my head: the Newsflesh series by Mira Grant heads each chapter with one or two blog posts from the characters. Mistborn and The Stormlight Archive by Brandon Sanderson open each chapter with quotations from in-world documents. And Dragonflight by Anne McCaffery opens each chapter with a verse from a song of the world.
They're popular because they can provide character voice and worldbuilding that don't fit in the main body of the story, but are still interesting and useful. Also, they can telegraph the themes or tone of a particular chapter, helping to guide the reader through the story. They can even provide foreshadowing, if the documents being quoted have information that the characters don't have access to. They can also provide bits of humor when the story needs it but it wouldn't fit perfectly into the story itself.
The big caveats to be aware of are that a) some readers skip epigraphs, so any information that is vital to the story should appear in the main story text as well, or you risk alienating those readers, and b) audiobook narrators don't always do a good job of separating the epigraphs from the text of the novel, and the transition can be confusing if the voices are too similar. If you think that there is a chance of your novel being converted into audio form, consider how your epigraphs will sound to that audience.
Neither of these caveats are meant to discourage you from including epigraphs (I personally am a huge fan of them), but are merely things to consider so that your epigraphs are the best that they can be.
I will agree with Lupus in that it can be incredibly useful for natural world-building and character development. I will also directly answer the question and say that it is hardly cliche, at least not on the same level as other cliches. It can even be done quite well, with the best examples I've seen being done as if telling the story as a foregone conclusion (i.e., "They dressed them up as pigs and sent them out to die." - History of the War).
The other question is to ask if they are needed. Including impressions of events discussed in your book would simply be repeating content, especially if your story would already be in first-person. Then, the style change doesn't serve a good purpose, and it could be dropped. However, if you include small moments from the characters past, that wouldn't come up in the course of normal conversation/flashbacks, then it would be encouraged.