How do I assign my email address to the Contact Us button?

You won't be assigning an email to your Contact Us button per se, instead, you should setup an Email Notification for your form.

Complete guide: Setting-Up-Email-Notifications

While you're at it, I strongly urge you to read the following guides as well:

How-to-Properly-Add-Change-and-Test-a-New-Email-Recipient-Address

Explanation-of-Email-Notification-and-Autoresponder-Settings