I would like to use my own domain name for my email address and use Gmail as my email client.
How can I do this without using email forwarding?
If you have an existing email account at your domain that you do not want to change, but you want to send mail using your Gmail account that says it is from your address at your domain, then you can just add a custom From address to your Gmail account.
In Gmail, go to Settings » Accounts » Add another email address you own to enter your address at your domain and send a verification email to that address. When the verification email is received at your other email address, click on the link in the email to verify that you own the address. Then it will be added to the list of From addresses that you can select when composing or replying to a message in Gmail.