Hi M.EContracting,
I would like to know which Office 365 plan you are using. When you say” How do I Migrate my custom domain /outlook emails to office 365”, do you mean that you want to move your custom domain and email data from Outlook.com to Office 365?
Have you moved your domain to Office 365? Get more info on how to Add your domain to Office 365.
After you have set up your custom domain in Office 365, email sent to your account holders will arrive in Office 365 instead of in Outlook.com. Existing email already delivered to Outlook.com will remain there.
However, you need to follow these steps to move email to Office 365. Set up Office 365 email account in Outlook 2010/2013 and then Use Outlook to move information between accounts follow the directions to export data from the desktop version of Outlook into Office 365.
For your reference see Help users move email and contacts into a new Office 365 for business account.
If you have any problem or getting error message during adding your custom domain/moving email data to Office 365, please provide us with the detailed information and we will be glad to further assist you.
Best regards,
Shyamal