There is no dedicated API for extensions. You can use the standard tools like the built-in devtools Timeline to profile the performance while developing and `window.performance` API to gather metrics during runtime.
The standard performance advices also apply e.g. import code on demand instead of using huge all-in-one bundles, this is particularly important for a content script that runs in all web pages. Chrome doesn't use code compilation cache for extension scripts so big scripts are recompiled in every tab thus slowing down page load process and draining the battery. Also, if the content script is only necessary after the user clicked your extension then don't declare it in `content_scripts` section, instead use chrome.scripting.executeScript to run it on demand.
There's one area to pay special attention now that ManifestV3 has removed the persistent background scripts: resource drain due to frequent restarting of the background script in case it observes frequent events like chrome.tabs.onUpdated or responds to messages from content scripts running in every tab, especially if the script is big and takes a long time to initialize the state. Chromium team hasn't yet acknowledged this problem exists, so you'll have to address it yourself e.g. rewrite the logic to initialize on demand, import code
on demand, maybe
prolong the lifetime of the background script to reduce the number of restarts during the day, maybe even offload some (or all) of the code to a web_accessible_resources iframe inside the web page.