Is this something that fits in a car trunk, or a garage?
When I was in high school, we built a rocket which lifted a nearly 2 kg payload about a mile high. That would have been about 4 kilometers range. It was fueld with zinc and sulfur and was 5 feet long and 2 inches in diameter. There are far more powerful fuels than we used, and a rocket the size of ours, or shorter and thicker would easily have the desired range and payload with easily obtainable fuel.
Depends on the fuel used. The shuttle's boosters have a specific impulse of 300 (one pound of fuel generates 300 pounds of thrust). That fuel is, IIRC, powdered aluminum and ammonium perchlorate. You can get up to about 450 or so with liquid hydrogen and oxygen, but that isn't practical for a weapon. Black powder will get you about 100-110, and the "smokeless powder" (actually a plastic compound) used in the WW2 bazooka will give you somewhere around 200 or so.
Now, you have to do a whole boatload of math to calculate how much fuel you need, because in addition to lifting the warhead, you have to lift the fuel you haven't used yet and the rocket carcass as well.
My vote on the fuel is some variant of smokeless powder.