use the gravitational pull of a massive asteroid, like Apophis or 1620 Geographos, and pull Mars off of its orbit, making it collide with Earth. This will create a giant impact crater, possibly ripping off a third of the Earth down to the mantle, and creating a shock wave that travels around the Earth, destroying everything in its path. After all, a Mars sized object did hit the Earth about 4 billion years ago, which mad the moon. It's not exactly blowing up the planet, but it will kill all life on Earth and destroy everything.
A good, but impossible, way to end life on Earth is to get Earth close to a massive dying star, or somehow make the sun blow up, this way when the sun or whatever star goes supernova we'll be close to it, and this will certainly rip the Earth Apart, Literally.
Here another good way to destroy the earth, and this one will actually be an explosion, a big one. ( of course this is pretty much impossible)
Antimatter - the most explosive substance possible - can be manufactured in small quantities using any large particle accelerator, but this will take preposterous amounts of time to produce the required amounts. If you can create the appropriate machinery, it may be possible to find or scrape together an approximately Earth-sized chunk of rock and simply to "flip" it all through a fourth spacial dimension, turning it all to antimatter at once.
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) is equivalent to the amount the Sun outputs in some 89 million years. Alternatively, if your matter-flipping machinery is a little more flexible, turn half the Earth into antimatter (say, the Western Hemisphere) and watch the fireworks.
Earth's final resting place: When matter and antimatter collide, they completely annihilate each other, leaving nothing but energy. All that would be left of Earth is a scintillating flash of light expanding across space forever. This method is one of the most permanent and total on this list, as the very matter which makes up the Earth ceases to exist, making it virtually impossible to even reassemble the planet afterwards.
Here another good way (again, impossible, just using my imagination an knowledge of astronomy)
Take every single atom on planet Earth and individually split each one down to become hydrogen and helium. Fissioning heavier elements to become hydrogen and helium is the opposite of the self-sustaining reaction that powers the Sun: it requires you to put energy in which is why the energy requirements here are so vast.
Earth's final resting place: While Jupiter, Saturn, Uranus and Neptune are gas giants composed primarily of hydrogen and helium, they are massive enough to actually hold on to their tenuous atmospheres. The Earth is not; the gases would dissipate away. You'd get a wispy mess of gas where there should have been a planet.
Here a good one:
Means for focusing a good few percent of the Sun's energy output directly on the Earth.
What I'm talking about here is: mirrors, and lots of them. Intercept several decent sized asteroids for raw materials and start cranking out kilometre-square sheets of lightweight reflective material (aluminised mylar, aluminium foil, nickel foil, iron foil or whatever you can scrape together). They need to be capable of changing focus direction at will because, while a few may be placed at the Earth-Sun system's Lagrangian points, the vast majority cannot be stationary in space and the relative positions of the Earth and Sun will be shifting as time passes, so attach a few maneuvering thrusters and a communications and navigation system to each sheet.
Preliminary calculations suggest you would need roughly two trillion square kilometres of mirror.
Command your focusing array to concentrate as much solar energy as you can directly on the Earth - perhaps on its core, perhaps at a point on its surface. So the theory goes, this will cause the Earth to generally increase in temperature until it completely boils away, becoming a gas cloud.