Google makes that simple to do in Chrome.
Open Gmail in Chrome. Log in to your Gmail account, if you haven't done that yet.
Chrome will (or should) remember that you are logged in to your Google account.
Set up a tab, or favorite for Gmail. That's not really necessary, as Chrome, being Google, has a quick link to any part of the Google "family" by clicking on the Apps icon.
Did I miss something about your question?