Summer Internship - A very well-known Professor
I want to let him know I will pay for my costs


This is not how things are supposed to work. It is normal that young researchers visit other institutions, but these visits are normally paid either by their current institution, or by the accepting one. What you're doing is explicitly proposing to do unpaid work, which is a bad practice and you don't want to work with someone who agrees to that.

What you can do instead is find 5-10 places where you'd like to go for an 'internship' or 'placement', etc and send emails with inquiries. It is fine to ask a professor whether they can hire you as an intern, but I don't think you should propose unreasonable arrangements.