Step 1: Go to your Google Site Edit page
Step 2: If this is a new google site select publish and skip to step 4, otherwise, Select the arrow next to the publish button on the top right of the page.
Step 3: Select Publish Settings
Step 4: In this menu look for the custom URL Section and select manage.
Step 5: Type in your owned domain name, an error message should appear underneath asking you to verify your ownership, select this. This will take you to a new page where you will have to verify your domain name.
Step 6: Use the provided drop down menu and find your domain name provider (Where you bought the domain name such as NameCheap.com)
Step 7: Once you select your provider a text string will pop up, copy this Text string as you will need it to verify your domain.
Step 8: Go to your domain providers website and login (NameCheap.com)
Step 9: Select manage next to the domain name you want to use for your google site.
Step 10: Select Advanced DNS.
Step 11: On this page you want to click add new record and select TXT Record from the drop down for the host use @ and then the Value is the text string you copied before next change the TTL to 30 min.
Step 12: Next add a new CNAME Record, for host use www and for value use ghs.googlehosted.com. Leave the TTL as automatic.
Step 13: Finally add a new URL Redirect Record, for host use @ and for value use http://www.YourDomainNameHere.TLD (The .TLD is whatever your top level domain is, in my case it will be .com), Leave this as unmasked.
Step 14: Make sure to save all changes to your DNS records if you haven't already.
Step 15: DNS records take around 30+ Minutes to change properly on a global level so now wait 30-60 minutes.
Step 16: After Waiting return to the Google verification page and select verify, if everything was done correctly it should work and you will be able to then publish your google site underneath your personal domain name.