The normal way to achieve this is for you to buy a custom domain, and then configure the DNS settings on your domain to point to NameOfCloudService.Location.cloudapp.asure.com
This part of the documentation shows how to do it with CNAMEs: https://video2.skills-academy.com/en-us/azure/cloud-services/cloud-services-custom-domain-name-portal#add-a-cname-record-for-your-custom-domain