i am self-funding, though so far still full time (living of savings). at my uni, if you will be self-funding, you simply go through the online application process (which involves a fee) and then get accepted or not.
a supervisor is assigned to you, usually in the offer letter, and then you decide if you want to take up the offer or not.
to accept the offer you will have to fill a financial undertaking form, demonstrating how you intend to pay for your course.