Moore is apparently not a mathematician by training. Here it is stated that he has a systems design engineering degree from Waterloo.
According to MathSciNet, the database that attempts to review every mathematical publication of interest, Moore has not published any mathematical papers, at least under the names "James P. Moore" or "J. P. Moore". The chances that an amateur without previous mathematical publications could solve these important and famous problems are, for all practical purposes, zero. (Prior to his celebrated recent success on the twin-prime problem, Yitang Zhang, a professional mathematician, had two published papers in good journals.)
Instead of placing his claimed solution on the arxiv, or publishing it in a journal -- as would be customary in such a case -- Moore is selling his solutions online in three different books for $27.05 each. One book is entitled either "The Proof of the Primes" or "The Proof of Primes", a title that doesn't make much sense mathematically.
Moore apparently is working with a public-relations firm to get the news out about his work. You can listen to an interview with him here; it is one of the most painful interviews I have ever heard, largely because the interviewer seems to have no comprehension at all about what the solution might consist of -- she keeps referring inexplicably to DNA -- and seeks to fill the time by repeating the same information over and over.
Moore gave a talk yesterday at the University of Waterloo, but I didn't attend. It wasn't sponsored by the Pure Mathematics department, though. As far as I can see, his public-relations firm hired the room. Again, that's not a good sign.
Here his PR firm suggests that his solution consists of "developing a formula capable of generating every prime number progressively and perfectly". This would not be of much interest, since such formulas are already known. The page also claims that such a method would "create stronger security systems". This is a common misunderstanding; encryption systems such as RSA, while they use prime numbers, would essentially be unaffected by faster ways to generate them. RSA's security would be affected by faster ways to factor products of two or more primes, which is a very different and essentially unrelated problem.
If amateurs think they have solved a famous problem, probably the best route to fame and fortune is to post the paper to a preprint archive. If you can't get an endorser for the arxiv, there's always vixra. Believe me, if your solution is correct, or even close to correct, you'll be acclaimed rather quickly. Hiring public-relations firms and selling your solution in books pretty much guarantees you will be ignored.
Addendum: here Mr. Moore claims, about the primes, that "there is no equation to define them". This is certainly false. They can be defined by a number of different equations; for example, see the talk by my colleague Eric Rowland here.
Another addendum, February 23 2014: Someone showed me a copy of Moore's claimed "proof" of Goldbach's conjecture. Needless to say, it is not correct, and introduces no new ideas at all.