How to use Lte in the Chrome extension (content.js) to draw a UI on top of current page?

I’m trying to build a Chrome extension that injects a Lit-based component into the DOM of any website. This component should appear as an overlay on top of the existing page content.
I’ve tried creating a Lit component and injecting it into the document body, but the component doesn’t render.

I’m looking for guidance on:

Best practices for creating overlay components with Lit.
Any specific considerations for using Lit in a Chrome extension environment.
Any code examples or suggestions would be greatly.