Here is my original tutorial on domain mapping with Go Daddy. Although if you need help with domain mapping and another registrant, just let me know.
https://www.blogsbyheather.com/2008/08/typepad-domain.html
In short, basically you are modifying the CNAME record (www).
Name: www (for example)
Points To: heatherporto.typepad.com (for example)
So, in addition to the above steps, while you are still in your Go Daddy (or other company you are using to register your domain name) account, add another CNAME record.
Name: blogsbyheather.com
Points To: heatherporto.typepad.com
So, like the original CNAME record you added where you pointed to your typepad blog, there you also point to your typepad blog but you do NOT have to do anything more on the Typepad side.
So, now if you go to https://www.blogsbyheather.com or https://blogsbyheather.com they both work and go to my typepad account.
Happy Blogging!