Likeliest - but alas already taken - answers
- anon's: "there is no easy way". I'd go for that one, even if it's anticlimactic.
- Ash's: "drill a hole in the magma chamber to trigger outgassing". By far the best "can do" answer, but it requires Yellowstone to already be in a close-to-eruption state (there must be enough dissolved gas in the right conditions).
- underground bombing: same problems in penetrating the magma chamber, plus that of having the bomb(s) - or as Carl Witthoft suggested, the antimatter containment unit - survive the trip.
Things that would not work
- Save drilling time using the Father of All Kinetic Impactors. There's probably a nice acronym in there, but the project just got mothballed due to lack of the required hyperdense penetrator. Unless we're talking of a Chicxulub event, which would put Yellowstone in the position of the tail wagging the dog.
- Laser drills. Same energy problem of the neutrino heater described below, only much larger. Also, extremely visible.
- Grey goo excavators. The little critters are tough, but the environment down there doesn't really lend itself to nanomachine work, what with those balmy 1500 °C and acid magma. This is actually one of the reasons why the computronium apocalypse is going to require some time.
Way more farfetched possibility
Our sociopathic billionaire wants Yellowstone to erupt even if it were naturally inclined not to. Now, volcanism is driven by heat rising out of Earth's core, and this heat is largely produced by the decay of heavy elements sunk in the lower mantle.
To increase the core heat directly below Yellowstone, one would need to increase the decay rate of at least the thorium chain of isotopes. There is some preliminary evidence that this might be possible after all.
So, at a ruinous cost, build enough efficient neutrino emitters (this is the most science-fiction-y part: we need an industrial strength neutrino emitter) almost wherever you want since neutrinos have no troubles in passing through thousands of kilometers of rock; and aim them on the volume directly below Yellowstone (for better results, the nearer you are, the more vertically you can aim, improving efficiency).
After sufficient time has passed (which depends on the neutrino flux, which depends on the energy and means available, thus on the money) the temperature beneath Yellowstone will have increased and a large "bubble" will be slowly slouching towards the surface. At the same time, surface temperature won't have changed detectably, reducing the possibility of spoiling the surprise.
At this point either let things proceed on their own, or accelerate the works with drilling or bombing.
Of course, radiation counters on the other side of the planet and neutrino detectors almost all over the world will start going crazy, which could be an important plot device - see whether a neutrino scientist can put the data together, triangulate and make the jump and deduce that someone is trying to cook off Yellowstone.
For the sociopathic billionaire, once the emitters are in existence through impressive feats of R&D, there will be the problem of how to camouflage them - hel'll need to set up plausible factories containing a mystery section where terrifying quantities of energy go in, and nothing much comes out (maybe he can try and market some side-effect of the emitter, possibly buying it himself under false identities).