IF your "People" contacts are all present and accounted for then, that would suggest that your Microsoft Live account (Hotmail, outlook, etc.,) is working.
If you configure both Live Mail and Outlook to use the same account then I would expect all your contacts to show up normally.
As a test, I had just recently performed a custom clean install of the latest Windows 10 preview build 14322, on which I set up the "Mail" app with my Microsoft Account that I use to log into that computer with.
Actually, I didn't set it up, it just used it automatically, all I had to do was click "all done" or "ready" or "let's go" or something similar. Checked my "People" app and all my contacts were there.
Subsequently I installed Office 365 (2016) using the same Microsoft account and all my contacts were there inside of Outlook.
Likewise I then installed just the mail portion of Windows Live Essentials from here
http://windows.microsoft.com/en-us/windows/essentials
Used the same Microsoft account and again, all my contacts were automatically displayed.
IF you have installed and configured Office Outook and LiveMail to use POP / SMTP rather than their default / automatic setup configuration, then you'll need to open a browser go to your Microsoft Account on the web (outlook.com, Hotmail.com, etc.,) open the people app from within the browser interface and select from the toolbar....
Manage -> Export for outlook.com and other services
Save the .csv (comma separated value) file to your computer and then read up on how to import that, into your version of outook and live mail.
Looks pretty easy, like in Live Mail -> Contacts
There is an Import Icon on the ribbon and .csv is one of the options, likewise in Office Outlook the Open & Export options can be accessed from the "File" menu
