Hi Orchids,
It depends really on the university, where you are up to in the application process, and whether you are applying for a PhD which has already been specified and funded, or whether you are looking for funding to undertake a project you have in mind. I am guessing that as you are talking about writing a proposal, you probably have your own project in mind, and are looking for somewhere to do it and someone to supervise it. You should probably ask the university itself about the application procedure, but I would think that you would need to write a letter stating your interest in research, those supervisors, that university etc, and also include some sort of written proposal. This probably wouldn't have to be in lots of detail, just a general idea of what you want to find out, why, and how you would like to go about it. Maybe just a couple of pages to start with. It might be best in the first instance to email the prospective supervisor about it if you have a particular person in mind, just to see if they would potentially be interested in supervising you and whether they are willing to support your application(s) for funding or have any available already.
Have you got a university and supervisor in mind etc, or are you just thinking about the application process at the moment?
Best wishes, KB