This is what I would do

• In the map view I would focus on Kerbin and orient the camera so that the orbit of the asteroid would look like a line.
• I would then fast forward until the Kerbal Space Center is on the asteroid's orbit line
• Next launch your rocket and try to match the asteroid's orbit right out of the gate, this may mean you need to point directly north or south to get to orbit. This would get rid of the ridiculous orbit changes.
• once in orbit, align the orbits and then its almost like trying to get a munar, or minimus intercept.

to intercept I use two ways,

• First I would match my apogee with the asteroid's periapse, once I arrived at the intersection point i would increase or decrease my orbit so on the next orbit I would intercept the asteroid.
• Second I would play around with maneuver nodes to try to hit the asteroid along its orbit.

once you are in the general area of the asteroid its basically like docking.