Meticulously and systematically deconstructed
You will need: a mass driver. A mass driver is a sort of oversized electromagnetic railgun, which was once proposed as a way of getting mined materials back from the Moon to Earth - basically, you just load it into the driver and fire it upwards in roughly the right direction. Your design should be powerful enough to hit escape velocity of 11 kilometres per second.
At a million tonnes of mass driven out of the Earth's gravity well per second, this would take 189,000,000 years. One mass driver would suffice, but ideally, lots (i.e. trillions) would be employed simultaneously. Alternatively you could use space elevators or conventional rockets.
Method: Basically, what we're going to do here is dig up the Earth, a big chunk at a time, and boost the whole lot of it into orbit. Yes. All six sextillion tonnes of it.
We will ignore atmospheric considerations. Compared with the extra energy needed to overcome air friction, it would be a relatively trivial step to completely burn away the Earth's atmosphere before beginning the process. Even with this done, however, this method would require a - let me emphasize this - titanic quantity of energy to carry out. Building a Dyson sphere ain't gonna cut it here. (Note: Actually, it would. But if you have the technology to build a Dyson sphere, why are you reading this?)
Earth's final resting place: Many tiny pieces, some dropped into the Sun, the remainder scattered across the rest of the Solar System.
Feasibility rating: 6/10. If we wanted to and were willing to devote resources to it, we could start this process RIGHT NOW. Indeed, what with all the gunk left in orbit, on the Moon and heading out into space, we already have done.
Source: this method arose when Joe Baldwin and I knocked our heads together by accident.