The fastest motorcycles don't really look like motorcycles at all, but more like missiles riding along the ground on two wheels. Even the engine powered LSR cars look a lot like that, for very good physics reasons.
At high speed, most of the force opposing you is air drag, which goes up as a square of the velocity. Since your vehicle is going to be moving just under the speed of sound at sea level, you will also be getting a lot of "wave drag" as well from the transonic flow (possibly some regions of supersonic airflow over parts of the vehicle as well).
So low frontal area is key (hence the missile shape), and the vehicle will also need to be made of strong but lightweight materials to deal with the forces on the vehicle, and have some clever aerodynamics to remain on the ground and under control at that speed. Air friction will also be an issue as well, so heat resistant materials are a must, and the interior of the vehicle will need a way to shield or reject the heat from the cockpit, and some clever engineering to ensure the waste heat from the engine(s) will also be able to be rejected.
Calculating the power required will need a lot more information (size, weight, acceleration, how far you have to travel at that speed), but engine powered LSR vehicles need a lot of horsepower (in the 1930's, it was common to use one or more 12 cylinder aircraft engines to power the cars to "just" 400mph (640kph), and one trick used then was to have the engines dump their heat into tanks filled with ice: using heat sinks rather than radiators, and dump the boiling water out at the end of the flying mile course and refill with ice at the other end). Most modern LSR's use multiple highly modified V-8 engines with forced induction (turbo or superchargers), as well as exotic fuel mixtures and power boosting tricks like water injection or nitrous oxide injection.
Your other requirements will be very difficult, if not impossible to do with the same vehicle. The traction requirements for an over snow vehicle are much different, and since snow is a loose, low friction substance, getting enough traction from the drive track will be difficult. LSR vehicles typically ride with minimal ground clearance to minimize some of the aerodynamic effects, and also have limited suspension travel and turning locks (you need to be pointed very accurately not only to hit the speed traps but also to prevent the aerodynamic forces from flipping the vehicle or turning you into a cloud of confetti), so cross country movement, hill climbing etc is out.
If you really need to go that fast, maybe you should consider an airplane...