Login to your Umbraco Cloud dashboard (https://www.s1.umbraco.io) and navigate to the project in question. In the top right click on "Settings" and then in the dropdown menu select "Manage Domains".
In the field provided add all of the hostnames for your new website.
Login to the control panel of where your domain names are registered - in this example we have used 123-reg although most providers will be similar. Navigate to the correct domain and then choose "Manage DNS (A, MX, CNAME, TXT, SRV)".
You should now set a CNAME record for your Umbraco Cloud given URL (e.g. website.s1.umbraco.io) and an A record for the Umbraco Cloud IP 184.108.40.206.
You should now be able to reach your Umbraco Cloud website via your chosen domain names.