I am able to self-fund a PhD, how can I use this when applying?
I may have an unusual perspective on this, since many mathematics Ph.D. programs in the U.S. fund all their students (at least through teaching). In this context, I'd recommend against even mentioning the possibility of self-funding:
It sounds unconfident, like you don't expect to receive funding, and this is not a positive impression to convey. You don't want to give the impression that you feel your own application is subpar, and that you're bargaining to try to turn a rejection into an unfunded acceptance. This might be a reasonable approach at the end of the process, if you're stuck on a waiting list, but it's a problematic way to begin.
Letting Ph.D. students go into debt feels exploitative, especially in a program in which most or all of the other students are funded, and letting wealthy people buy their way into graduate school is also troubling.
There will be worries about how sustainable your self-funding is. In a year or two, your money or willingness to go into debt may run out, and the department would have to choose whether to come up with money for you or kick you out. That's an unpleasant choice, and the admissions committee may even worry that you are deliberately aiming for this situation (in the hope that they won't be willing to kick you out once you've joined the department).
In the sort of programs I'm familiar with, announcing in advance that you would be willing to self-fund a Ph.D. won't increase your chances of admission. I've only seen this come up very rarely, and the admissions committee has simply ignored the offer of self-funding and made a decision of acceptance with funding or rejection. I can easily imagine that it could decrease the chances of admission a little, although I haven't seen a clear-cut case of that.
By contrast, if you are awarded a prestigious fellowship with guaranteed funding for a certain number of years, you should certainly make sure admissions committees know about it. That doesn't have any of these disadvantages. However, it's not in your best interests to offer to pay your own way unless that's a common and respected approach in your field.
To add to Anonymous Mathematician's answer (I'm also in math), if you get an external fellowship, this can certainly increase your chance of admission. When I applied, I initially got waitlisted by one top univerisity, then accepted when they heard I got an NSF Graduate Fellowship (in fact, the way I learned I got the NSF fellowship was from my acceptance letter from this university, which incidentally I didn't go to).
In the cases of self-funding, sometimes it does help, if you're strong enough to warrant acceptance with funding, but the department is low on funding that year. However, in most cases, I agree it is not so helpful to advertise this up front, particularly with top programs, but if you get waitlisted and seem to be a borderline case, then letting the admissions committee know you would be interested in admission even without funding may help.
Where I think it will be more helpful is, if you have trouble getting into a good PhD program, you can use self-funding to help get yourself into a masters program first (after which, if you do well, will have better chances for a good PhD program). Some departments will fund masters students, and some not. Ours typically does, but definitely accept some applicants we would not otherwise, just because the have self-funding. My guess is this may be rather common, as master's degrees are not as serious commitments as PhD's, and one worries less about the quality of grad students getting watered down in this way.
Given your position, I would suggest approaching professors directly that you would like to work with to discuss what possibilities there may be. A student who can fund themselves should have more freedom in terms of choosing what they want their research to focus on. This may be interesting for professors that have some ideas for projects they have no funding for but would like to investigate. It may also let you choose your own research program and get a professor to agree to supervise you with it.