You don’t need the services of the mail any longer or it’s probably outdated then take the below steps to delete your gmail account.
Sign in to your gmail account at gmail.com on the web.
Click on the account preferences page (this can be seen at the top right corner of your account) just click on your avatar or your picture (or profile) and a drop down menu will appear which would consists of log out button and others, there you would find the account settings.
Click Delete products. You may need to sign in to your account again.
Click the delete button that is close to gmail.
Follow the instructions on the screen, then choose Remove Gmail.