However, with this new update, people who use their personal Google account to access Google Classroom can only join classes. The ability to create a class will be available some time in the future. So if you are interested in using Google Classroom, I would recommend signing up for G Suite for Education.
To use your personal Google account for Google Classroom simply sign into your account, and go to Google Classroom, click the + sign to join a class. Note, you must know the class code to join an existing class.
Additionally, the Google administrator in your organization will need to enable settings in the admin console to accommodate users to have access to Google Classroom. They possibly will need to whitelist domains and enable class memberships. Click here to read more about the settings on the admin console. See the animation below to navigate to these settings in the admin console.
Click here to read more about this update.
A new update as just been release allowing users with personal account to create classes as well. Click here to read more about it.