First of all please log in on the Office 365 Home Portal. Click the Domain link which is located under the Management section and then Add new domain link.
Now please specify your domain, check it and then click Next.
Now it becomes a little bit tricky for the non-technical user. Microsoft wants to verify that you own the domain so you have to create a TXT record at your domain registrar. From my experience I can tell that after you do the update you have to wait between 2-24 hours for the change to be propagated on the internet.
After the verification is done you can add your services. Please choose only Exchange Online and Lync Online at this stage. We will add the SharePoint Online public web site later.
Then to activate this services you have to change the DNS Records at your domain registrar.
Trackback from your site.