Connect domain

So let's say I have a client so I build the website on my account because I'm not giving it over till they pay . They buy the domain on their account how would I connect the domain to the website how do i get access to the domain?

You would want to give them access to the website and they can then assign the domain to the website that you've designed.

Shouldn't it be my job to connect the domain and how do they assign it?

You should have them purchase the domain via the new website, but you also get a free domain for a year with a new business plan, so you could also just use that for the first year.