If by “installed from my .crx file” you mean installed from Chrome Web Store you can simply check extension manifest.json for value of update_url attribute. CWS adds it when you upload your extension.
If you have a self-hosted .crx file, get your extension information using chrome.management.getSelf() and check installType of returned ExtensionInfo object. If it says "development" that means that extension was loaded unpacked in developer mode. "normal" means that it was installed from .crx file.