Is it possible to determine which user profile is logged in Chrome from my custom extension using Chrome API?
There is no Chrome-API for that, the best (any only) way is to use oAuth:
Yesterday Google has published news about the Identity API at the Chromium blog.
The Identity API may only work with packaged Apps!
The Identity API allows packaged apps to authenticate users securely using OAuth 2.0 without the user having to provide a username and password directly to the app. The Identity API supports authentication using Google Accounts as well as third party providers such as GitHub and Foursquare.
The Identity API also gives packaged apps secure access to Google APIs such as Google+, Calendar and Drive. As an example, Google Keep uses the Identity API to authenticate users as well as to call the Google Drive API to save notes to Drive. The Identity API uses a webview based UI to show the OAuth consent dialog and when the Google+ API scope is used, it allows users to control who can see their activity on the app.
Documentation at developer.chrome.com: https://developer.chrome.com/trunk/apps/app_identity.html
If it's possible to do so, consider writing a Chrome packaged app instead. The Identity API will give you the information you need.