Hi Amye,
You can publish your onto Business Catalyst by simply opening your muse file and selecting Files->Publish .
Please refer to this tutorial :- Publish your site with Adobe Web Hosting | Learn Muse | Adobe TV
Also, once your site is published, then you can add your domain to your site hosted on Business catalyst.
Please refer to this tutorial in order to add your domain :- http://helpx.adobe.com/business-catalyst/using/change-site-domain.html
Hope this helps
Regards,
Rohit Nair