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Unfortunately, that information is very old and outdated. It's no longer valid and following the advice on these sites will only try your patience at a time when you don't need more stress.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 Accessing Outlook.com, OneDrive and other Microsoft services when someone has died.
The short version: If you want to close the account and know the deceased person's password, you can close the account yourself. If you need to close it and don't know the password, it will automatically close after 2 years of no access. If there is a subscription on the account, it will cease if the payment does not go through because the bank account or credit card is closed.
If you need to access the account and don't have the password, you will need to contact Microsoft.
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.