The main tech you need is the claw; you use it to grab the asteroid for pushing:
As long as you've got that, yes, you can definitely divert an asteroid from hitting Kerbin, though as you've realized, you'll need to rendezvous before its periapsis.
Build a spacecraft with a claw, a Poodle, and maybe 3000 m/s delta V, give or take
Attach a launch vehicle that can put it into orbit (i.e., so the 3000 m/s starts once you're in orbit)
Launch it to the same inclination as the asteroid
Target the asteroid
Place a maneuver opposite the asteroid's incoming path
Increase the maneuver's prograde component and watch what happens with the close approach markers
Fiddle with the maneuver; eventually you should be able to put your craft into the asteroid's path just as the asteroid is reaching the same spot; that's your intercept!
Rendezvous with the asteroid (see various docking tutorials for help, this is its own process)
Open the claw and grab the asteroid
Rotate so you're pointing perpendicular to the asteroid's path
Burn and watch the asteroid's periapsis ascend out of the ground
As soon as it's >70 km, you're done!
EDIT: Further note, 15 days might be a little too long to do this conveniently. It only takes about 7 days to get all the way out to Minmus, which is nearly at the edge of Kerbin's sphere of influence, so if this Kerbin-killer is 15 days out, you might be looking at an intercept outside of Kerbin's SOI, which can be harder to plan. If you wait till it enters Kerbin's SOI, getting the intercept should be a sure thing.