TL;DR: Finally, my years of experience in Kerbal Space Program pay off. It's possible to do, but won't do much compared to just blazing through the atmosphere.
As @TimB mentions, you can't "skip" an asteroid from the atmosphere quite the same way as you would skip something off the surface of a pond. You can, however, make use of the atmosphere to change your trajectory and effect a "bounce".
To get an idea of how this effect would work, let's forget about planets and orbits and work for the moment with an infinite flat plane with a homogeneous gravitational field. Even if you fire it of with a lot of horizontal velocity, it will eventually just drop. Add atmosphere and it just gets slightly slowed down towards the end.
If instead of an asteroid you use a glider, however, you have options. The most straightforward one is to pull up, which will under favourable circumstances get you back out of the atmosphere again, but with less velocity. You may be able to repeat this process, but eventually you'll just glide down. We can do something similar with an appropriately shaped asteroid.
To planets now. We won't be doing any "slingshots" or "gravity assists", since those don't actually require an atmosphere and don't work in two-body systems anyhow.
What happens when a spherical asteroid passes sufficiently close to a planet to dip into the atmosphere (but not so close as to hit the planet) is that it slows down some, losing velocity (and thus energy) to aerodynamic drag. This changes its orbit; doing it on purpose is called "aerobraking" and if the orbit goes from hyperbolic (ie. speeding back into space) to elliptic, it's called "aerocapture". It's not, strictly speaking, "skipping off", since it's not the atmosphere bouncing you off, it's orbital mechanics carrying you away.
Now the thing to understand is that your orbit is at any given point fully determined by your position relative to the body you're orbiting, and your velocity relative to that same body. Aerobraking changes your velocity, generally just by braking, which has the effect of shortening your semi-major axis (bringing you to a "lower orbit") and bringing your periapse down some, as both horizontal velocity and vertical velocity are affected equally.
At this point, using wings (and the golden rule of aircraft design tells us that at these velocities, anything is a wing) you can cause the drag to be asymmetric, gaining what we call "lift". Note that it's impossible for you to gain energy this way, the only thing that's happening is that you're trading some of your velocity to change the direction of the rest of it. You can take advantage of this to change the altitude of your periapse (and hence the eccentricity of your orbit) or effect a plane change, but you'll lose energy doing so.
If you have a controllable aerodynamic shape, you can take advantage of a planets atmosphere to alter your trajectory at no cost in propellant to you. Apollo capsules (IIRC) took advantage of this; by having a centre of mass slightly offset to the side from the geometric centre of the capsule, they would get carried away slightly to the side in an atmosphere, and could rotate the capsule lengthwise to gain some limited control authority.
I have taken advantage of this trick to keep the periapse of a moon lander in the high atmosphere during multiple aerobraking passes on a return from a moon, allowing me to gently slow down and rendezvous with a space station in orbit of Kerbin at a minimal cost in propellant.
Could you use this as a weapon? Maybe. The energy expended in these maneuvers manifests as shock heating and is usually absorbed by the orbiting body, so the effect on the planet is limited. But what if there was enough of it?
This begs the question of how much energy this maneuvre consumes. We can get the answer from the vis-viva energy equation. Taking the initial orbit and calculating the energy (multiplied by the mass of the asteroid), we get the maximum energy we may deposit by deorbiting (read: crashing into the planet). The difference between the energy of this orbit and the new orbit is how much energy was expended/deposited.
For Earth-like planets, just attaining escape velocity compared to sitting on the surface gives you an energy $-62,6 MJ/kg$ which is about $15x$ the energy of TNT. Impressive at first glance, but not much in the grand scheme of things, especially if you consider that you're only expending a tiny part of this energy (however much you're willing to sacrifice without falling) an most of it will be absorbed by the asteroid.
Perhaps a better use of this capability would be to just drop some kinetic impactors (like tungsten rods) to do some damage and then use the planetary atmosphere to fine-tune your trajectory to your next target.
Oh, and since you were asking about the shape of the asteroid: it would probably end up being vaguely reminiscent of a space shuttle if you wanted to optimize, but unless you dip too deep, any shape with asymmetric drag or control surfaces would do, albeit with less efficiency.