Your literature review gives your readers an understanding of the evolution of scholarly research on your topic.
In your literature review you will:
Throughout the literature review, your emphasis should fall on the current scholarly conversation. This is why the rubric often specifies that you need resources from peer-reviewed journals, published within the last five years of your anticipated graduation date. It's in these recent, peer-reviewed journals that the scholarly debate is being carried out!
The literature review also shows the "gap" in the conversation -- and how your own doctoral study will fill that gap and contribute to the scholarly knowledge. This is where you make the case for the importance and usefulness for your own work.
To become an expert you will ground yourself in your topic by reading very broadly; you will read a wide variety of sources, not just peer-reviewed.
Once you have a fuller understanding of your topic and the issues involved, you can more successfully search for scholarly resources on the topic, and synthesize them in your literature review.
Searching comprehensively is important. You need to be confident that you have found all of the relevant materials, so that you have a full picture of the topic and can share that picture with your reader.