I have been running SF on Azure websites since 6.x
Easiest way is to download the website publishing file and publish the site to your azure website.
After the initial deploy I exclude the Configuration folder but its up to you to decide how to manage that.
I also use the web.config transforms to handle the differences between my local and azure instance. I also move the database connection string to the web config file.
You also need to enable the Telerik element and tell that the install is on Azure. (Use the web transform for this.
You probably can run it on the shared Azure websites but SF needs a lot of memory so you should plan to use a dedicated instance.
One issue you may come across is 'Runtime error..can't find msshrtmi.dll
Basically this is a reference from the Azure files in the project.
Fix Find the dll on your PC and add it to your project. I use a binding redirect to keep the error away.
<assemblyIdentity name="msshrtmi" publicKeyToken="31bf3856ad364e35" culture="neutral" />
<bindingRedirect oldVersion="0.0.0.0-188.8.131.52" newVersion="184.108.40.206" />
And I use Thunder to deploy the database, (initially).
That's the main things off the top of my head.