I just spoke with an awesome agent who should me how to change the email for your Microsoft account when you've lost access to the original email!
Now, the website to do this at is account.live.com. Usually you can't change the primary alias without having verifying it first. The agent told me that there is actually a "back door" to this. BE WARNED, you wont be able to do this option if you have money on your Microsoft account.
Step 1: Sign into account.live.com with the email and password to your Microsoft account
Step 2: When on the account summary, you'll see a message saying the following "Your Email address has not been verified" followed by a blue hyperlink stating Verify email address. Click on that link.
Step 3: After you click on the link, you be on a page that'll tell you that a verification email was sent to that email address. Ignore this page and open a new tab (Internet explorer would be preferred)
Step 4: On the new tab, go to www.outlook.com
Step 5: After going to www.outlook.com, you will see a blue link saying "Change the email for your Microsoft account". Again, you won't see this link if you have money on your Microsoft account.
Step 6: Once your on the "Alias Management" Page, use the option for an existing email for the primary alias. The Primary alias for the Micorosft has to be an email that you've never used with Microsoft/Xbox. The website will give you an error.
I would recommend talking to an agent at 1-800-469-9269 to further assist you, You can also PM me if have any questions. NOT EVERY AGENT WILL KNOW THIS OPTION.
HAVING YOUR EMAIL VERIFIED IS NEEDED TO DOWNLOAD YOUR PROFILE ONTO THE XBOX ONE! Also, you want to make sure that your security information and updated with at least 2 alternate pieces of contact information.