On BrowserStack Live, you can easily install Chrome extensions on a remote Chrome browser by launching a session and installing them through the Chrome web store, similar to how you would do on a regular Chrome browser.
If you wish to test the local build of the Chrome extension then you will need to first upload the CRX file on any file sharing tool, and get the link of the same. Next, launch a session on BrowserStack 'Live' and access the link within the session and download the CRX file on the remote machine.
Once done, you can install the extension by going to chrome://extensions on the remote Chrome browser and dragging/dropping the CRX file. Please watch a short demo of the same here.