Let us begin by expressing our condolences for your loss.
Outlook.com, Hotmail, Live, and MSN e-mail accounts are treated on one Web site.
Microsoft recently changed its website, invalidating older links. Currently, the information is available at: "Next-of-kin process for Outlook.com" (previously available at http://windows.microsoft.com/en-us/outlook/next-of-kin, which, however, is now on a website where the information is not displayed).
Other references show similar advice: "I have to go into a deceased personal account" or "How to get in touch with various digital services".
You must contact Microsoft's Custodian of Records. The information provided on their website is this:
If a family member dies or is medically disabled we are here to help. We have a dedicated team that can share Outlook.com content that belongs to the family member, including all emails and their attachments, address books and contact lists, for a relative or guardian.
To protect the privacy and security of all Outlook.com customers, we can not provide you with the password, change the account, or transfer ownership of the account. We also can not assist you with password reset, account recovery, or other support for your own account.
The content will be transferred to a DVD that will be sent to you. E-mail the Microsoft Custodian of Records to request content sharing and close the family member's account email@example.com,
Please enter the following information in the e-mail:
Your first and last name
Your phone number.
Your e-mail address.
The full name of the deceased or disabled family member.
To the best of your knowledge, the family member's Outlook.com accounts.
Our team will contact you with details and specific documentation requirements.
If you have the family member's credentials (such as your email address and password) and you can sign in to the account or accounts, you do not have to go through the process of the next of kin. You already have access to all the information we would put together on the DVD.
Although the reference to outlook.com appears to refer to, if you visit https://www.hotmail.com/ and use the "Contact Us" link in the footer, you will end up here.
A Similar Question Signing up for a deceased Facebook Facebook account for Facebook users has been posted on SE's WebApps website (here). For Stack Exchange users, we have the following questions and answers about Meta: How to handle the death of a user? It's fine to ask this question here, but you need to contact Microsoft for further assistance.