See the solution from Microsoft here: http://windows.microsoft.com/en-US/windows-live/account-close-account
The relevant section is as follows:
To close your account
- Go to the Microsoft account overview webpage, and then sign in with your Microsoft account.
- Click Close account.
- Verify that the correct Microsoft account appears, type your password, and then click Next.
- If you have one or more aliases associated with your Microsoft account, you'll be prompted to remove them. If you have a Hotmail account, click deactivate your Hotmail account.
- or -
If you're closing your Outlook.com account, choose your reason for closing your account, and then click Deactivate account.
To close only your Outlook.com account
Your Outlook.com account will become inactive if you don't sign in for more than 365 days or within the first 10 days after signing up for your account. After an account becomes inactive, all messages, folders, and contacts are deleted, and incoming messages are sent back to the sender as undeliverable. If you want to close your account right away:
- Visit the close account webpage.
- Choose your reason for closing your account, and then click Deactivate account.