The same way as connecting to any other database! Personally, I use EntityFramework and create my own service. You can use EF and use RIA Services to create a lot of code for you though if you prefer.
Add an Entity Model to your project, create from existing db, point at your Azure Db, job done. Remember to add your existing IP to the Azure firewall while developing.
EDIT: Try following this tutorial instead: http://msdn.microsoft.com/en-us/library/ee707376(v=vs.91).aspx BUT one thing to watch out for. When you come to the step to add a DomainService, if it hasn't picked up your context you will need to follow the steps listed here: http://support.microsoft.com/kb/2745294 ... which are easy to follow. Delete the TT files and change the model code generation strategy from none to default.