There are few things you will need to change in your Godaddy account before setting custom domain in Windows Azure. They are:
Point CNAME to Azure Website
Change CNAME for www subdomain to point to Windows Azure website.
In Godaddy’s Domain management portal, click on “Launch” link for DNS Manager.
Under CNAME category, change WWW host to your Windows Azure website.
After saving the zone file, it will take about 1 hour for the DNS to propagate across. So, you might not be able to do next step immediately.
Forward Domain to WWW Subdomain
Still on Godaddy domain management portal, forward your host domain to WWW subdomain.
Custom Domain in Windows Azure
In Azure management portal, configure website which you want to set custom domain, click on Manage Domains button on bottom bar.
After DNS change has propagated, you can now put in your Godaddy domain name under Domain Name field.
How Windows Azure Website Works
How Windows Azure website works has to do with why you did what you just did.
Every time you create a new website, Windows Azure assigns a host name for your site under azurewebsites.net (mysite.azurewebsites.net). One of the reason for this is, if the IP address ever change for mysite.azurewebsites.net, for example: geolocation purpose, you don’t need to change anything from your side, so you will still be using azurewebsites.net host name you are given.
To point your custom domain, mysite.com, to mysite.azurewebsites.net, you will need to use CNAME record. There is another popular way to point a domain to a server, it’s called A record, however, we can’t use this in our case because A record only point to IP address. And since you have pointed WWW subdomain to your Azure websites, you will also need to forward mysite.com to http://www.mysite.com.