If you don't already have one generate an SSH key pair.
Go to the Dokku on Azure deployment page and click Deploy to Azure.
You'll be prompted to enter a few parameters, including a unique storage account name and a unique name for the subdomain used for your public IP address. For the sshKeyData parameter, copy and paste the contents of the public key file you just created. After a few minutes the Dokku instance will be deployed.
In your browser of choice, navigate to http://[dnsNameForPublicIP].[location].cloudapp.azure.com. Where [dnsNameForPublicIP] and [location] are template parameters you used to deploy the template.
Finish your Dokku setup like you normally would by creating a new public/private key pair for your deployments using ssh-keygen (don't use the same one as you created in the first step). You should select Use Virtual Host Naming and set the Hostname to a public DNS name that you own such as one you would purchase from Namecheap. Alternatively thanks to xip.io you can just use [yourAzurePublicIP].xip.io for free. For example, if your public IP is 44.44.44.44 then you would set it to 44.44.44.44.xip.io.