It's very simple, actually. First, log in to your Gmail account and choose (upper right corner) Settings > Accounts. Click "Add another email address" and follow the steps to add the address you want to use. This will allow you to send mail from that address through Gmail.

Then, log in to your Cpanel, go to your webmail and and use the forwarding option to forward mail to your primary Gmail address.

(For instance, on one of my accounts, my primary address is daysofblunder ******.com. I receive mail from other email accounts such as scripts@daysofblunder.com by forwarding them to this address and I'm configured to send mail from that address through my Gmail account.)

The only time I have to log in to my webmail account these days is to add or remove an email address.