I wouldn't sign for anything that involves certainty of death.
Everyone dies sometime...
I'm sure any governmental support for a one way trip is out of the question. Far to many layers of far to many problems to even waste the time contemplating it.
However, it is entirely plausible for an eccentric billionaire to forgo trying to take over the world and would simply try to "get away from it all" in a dramatic fashion.
National space agencies are horribly inefficient with time and money. They are far to busy getting everything signed in triplicate, dotting their eyes, crossing their tees, and basically covering their bureaucratic asses, that its as likely that a upstart private venture reaches Mars before a government does.
I think its possible to be able to put together a craft and launch it for a tenth of what NASA or ESA does. With the ease of knowledge access through the internet, the wide spread and cheap availability of rapid, precision engineering and fabrication, hardware can be built up much faster than possible before. Reach an agreement with a nation that has aerospace aspirations, (Brazil, India, Israel, Iran, etc.) for an assembly and launch site.
I'm sure the project and journey will be a media circus, that is hard to avoid these days. But it CAN be portrayed as something other than as a ghoulish spectacle of watching someone travel to their demise. It would be worthy of doing even discounting the vanity of the historical entry that it would earn. The trip could render alot of good data and even lessons about surviving in space that will have to be learned the hard way, no matter who it is. Better to send a single "canary in a coal mine" than an entire crew of the planet's best and brightest.