Applying for grad school without undergraduate background
I agree with the other answers — admitting a student without formal academic background is very risky.

As a first step, I suggest contacting graduate programs asking if they allow non-degree students to take their classes.  If so, taking classes as a non-degree student will let you build up a formal academic background and strong recommendation letters in your chosen field, making your case for admission much stronger.  And if you're admitted to the PhD program at the same university, you may be able to use the classes you've already taken toward your degree.

Taking classes will also give you a sanity check on your self-assessment.  From personal experience, it can be really hard to spot gaps in the knowledge you build from self-study, simply because you don't know enough to ask the right questions.

But once you get your foot in the door, do not just take classes; do whatever you can to get involved in research!

One significant downside to non-degree classes is that you'll almost certainly have to pay for them out of your own pocket.