Since virtual machines on Azure are hosted a little bit differently you have a public virtual IP address known as a VIP. You then get an internal IP address which is what your virtual machine will see when setting up your bindings on IIS.

You can see your public & private IP addresses on the azure portal dashboard under the quick glance section. So when you set up your DNS records use the VIP.

Then in your portal for Azure set up a TCP endpoint listening on port 80.

Should be working for you now