Blown up
You will need: 25,000,000,000,000 tonnes of antimatter.
Method: This method involves detonating a bomb so big that it blasts the Earth to pieces.
This, to say the least, requires a big bomb. All the explosives mankind has ever created, nuclear or non-, gathered together and detonated simultaneously, would make a significant crater and wreck the planet's ecosystem, but barely scratch the surface of the planet. There is evidence that in the past, asteroids have hit the Earth with the explosive yield of five billion Hiroshima bombs - and such evidence is difficult to find. It is, in short, insanely difficult to significantly alter the Earth's structure with explosives. This is not to mention the gravity problem. Just because you blasted the Earth apart doesn't mean you blasted it apart for good. If you don't blast it hard enough, the pieces will fall back together again under mutual gravitational attraction, and Earth, like the liquid metal Terminator, will reform from its shattered shards. You have to blow the Earth up hard enough to overcome that attraction.
How hard is that?
If you do the lengthy calculations you find that to liberate that much energy is equivalent to the complete annihilation of around 1,246,400,000,000 tonnes of antimatter. That's assuming zero energy loss to heat, neutrinos and radiation, which is unlikely to be the case in reality: You'll probably need to up the dose by at least a factor of twenty. Once you've generated your antimatter, probably in space, just launch it en masse towards Earth. The resulting release of energy (obeying Einstein's famous mass-energy equation, E=mc2) should be sufficient to split the Earth into a thousand pieces.
Greg Bear's novel, "The Forge Of God", contains an interesting refinement of this technique. Here, the antagonist instead generates antimatter in the form of a "slug" of anti-neutronium - superdense material massing a billion kilograms per cubic centimetre. This is fired into the Earth's core. Neutronium passes through ordinary matter as easily as a ball flies through the air, so the anti-neutronium slug doesn't annihilate immediately; rather, it builds up a protective sheath of plasma around it as it plunges downwards towards the Earth's core. It's then followed up by a slug of regular neutronium, which also falls into the core, at a time calculated to meet the first slug head-on at the exact centre of the Earth, where they annihilate themselves, and soon afterwards, the Earth itself. Highly space-efficient, and with the added bonus of all the energy being released at the Earth's core, where it can do the most damage. In the book, the antagonists simultaneously detonate nuclear warheads in certain oceanic trenches, to weaken the crust and allow the planet to be blown apart more easily.
Rearranging Earth into two planets - which, provisionally, is sufficient according to my current criteria - would take slightly less energy, but considerably more finesse.
Earth's final resting place: A second asteroid belt around the Sun.
Comments: trembling writes, "I still think that antimatter is crazy s**t, i.e. wouldn't want it on my flapjacks". Charles MacGee presents a very well-realised alternate source of explosives in his blog; this method involves generating the explosive energy by fusing together the lighter elements of Earth's mantle (magnesium and oxygen). Of course, this would involve the invention of an efficient magnesium fusion bomb. And then turning all of the Earth's mantle into bombs. How implausible! Well. Implausibility is a relative thing.
Getting easier.
Feasibility rating: 4/10. Just about slightly possible.