To attach your custom domain to the Outlook.com servers, you need three things:
Your own domain, which you can purchase from any registrar.
The ability to create custom DNS (Domain Name System) records. In particular, you’ll need to create an MX (mail exchanger) record.
A Microsoft account (formerly known as a Windows Live ID), which you’ll use to sign in to the domain administration center. This account should use an address that is not on the same custom domain as the one you're about to attach to Outlook.com. (If you already have a Microsoft account you use for other purposes, use it here.
If you're a stickler for legal details, you can read the terms of service, which are a supplement to the Microsoft Services Agreement.