The big question is, how much will it cost? The estimates that I've seen indicate it would cost about \$560 million per launch, assuming no re-usability. If that is all it takes to get to Mars, then Elon Musk can get the money fairly easily, from a number of sources. Prior to launching humans to Mars, they will have launched one cargo mission unmanned, several LEO ranged tests, done R&D, and at least 3, maybe more, Red Dragon Missions (Estimated around \$150 million, on the low side). Put all of that together, and there's easily a \$2 billion cost in non-R&D costs. R&D is likely on the order of \$3 billion, so that's around \$5 billion minimum.
Of some question is how much they can sell of the initial runs to space. I can imagine someone paying for cargo launched to Mars at very low cost, for instance. Get the price low enough, and many people would have payloads they are interested to sending to Mars on a Red Dragon. NASA might even pay for loads taken there, for instance, and they might even completely make up the cost of the mission for the subsequent missions.
Elon has stated that his potential funding sources include "Stealing underpants, Launch Satellites, Send Cargo to the ISS, Kickstarter, and Profit". More seriously, he committed to using his life's fortune to make this happen.
Will NASA fund it? Maybe. I can't imagine that NASA would turn down an opportunity to be on that first spacecraft to Mars that SpaceX has, and I can't imagine SpaceX refusing to give them that opportunity. The first missions will be far more than the \$200,000 end goal, but let's say that they assume no reusability for the first launches, and the \$560 million price tag holds. That would lead to a cost of only \$5.6 million! NASA is already going to pay SpaceX around \$25 million per seat to the ISS, possibly more. I can't imagine that if they meet the required safety margin, NASA wouldn't book seats on the first SpaceX manned mission to Mars. They might even pay a premium to be the first out of the hatch;-) But I don't think they will pay for SpaceX to do the R&D, at least, not directly.