How to bring up your list of accomplishments in performance review?

It seems like your performance review is more ad-hoc and unstructured than a formal 360 feedback system like at a Big Tech company, so I assume you can't just work it into your self-review and be mostly done with it.

That being said, here are my thoughts:

• Share your brag journal with your manager - In general, you should be sharing it in real-time with your manager during your 1 on 1 meetings: It's important to regularly let your manager know your wins as managers are very busy people and can easily miss things. Since performance review is coming up, just share the entire document with them ASAP. You can send it to them via a chat message so they can "soak it in", and then have a discussion with your manager in your next 1 on 1 about it and how this helps you be represented well in performance review.
• Work backwards from your expectations - If you have levels at your company, working backwards from the expectations of the next level helps as well. The idea is to use the accomplishments from your Brag Journal to "check the boxes" of these expectations. Let's say the expectations of the next level for you are "Lead a team of 2-3 software engineers." And across the past performance cycle, you have consistently led a team of 3 engineers to ship multiple projects. You can use this to objectively make the case that you should be promoted to the next level. You can write this entire "case" into a document (which would effectively be a self-review).