There are a number of articles on the web with instructions on what you need to do to access a deceased family member's Microsoft Outlook.com account. The instructions include contact information that is not longer in user and instructions to send copies of the death certificate by email or fax.
Unfortunately, that information is very old and outdated. It's no longer valid and following the advice on these sites will only try your patience at a time when you don't need more stress. You need to go straight to the source: Microsoft.
Microsoft's policy and instructions are at Next-of-kin process for Outlook.com.
The short version: Microsoft will not respond to faxed or mailed requests. Microsoft must be formally served with a valid subpoena or court order before considering a request to release a deceased or incapacitated user’s information contained in an Outlook.com email account. Microsoft will only respond to non-criminal subpoenas and court orders served on Microsoft’s registered agent in the requesting party’s state or region.
To find the registered agent for your state or country, check with the government in your area. You may be able to get the registered agents address from Microsoft support. For a list of regional phone numbers, see Global Customer Service phone numbers.