If it was in the sky, meaning in the atmosphere, it would burn up like any high speed object would. If it were large enough, pieces of it, maybe very large chunks, would make it to the ground and create a crater, just like any other very large piece of matter.
If it was above the atmosphere, it would have to be firing some very large rockets in order to maintain its course at a constant altitude in order to continue "whizzing around the Earth at near light speeds". Normally objects in Earth orbit are just coasting at a speed determined by their altitude, the higher the altitude, the slower they go, the fastest speed just above the atmosphere where they won't burn up is less than 20 thousand miles per hour, no where near close to the speed of light.
So, assuming that this craft is just above the atmosphere and firing rockets to maintain its near light speed coarse, it would make about seven "orbits" per second and unless it was changing its altitude or position of its "orbit", it would likely be caught in the afterburn of its own rockets and be destroyed.
But assuming that the aliens on board are just smart enough to avoid this problem and to steer their craft on a course where they don't get caught in their own afterburn, then we might be able to see the exhaust of their rockets as it pollutes the space surrounding the earth. You must not think that the exhaust from a rocket in space looks anything like it does from a rocket in air. In air, the exhaust takes time to dissipate so it forms a well defined trail behind the rocket. In space, it spreads out in all directions very quickly so it will be very difficult for the aliens to avoid. These aliens have to be very smart in order to construct a craft to get all the way to Earth and to steer it to avoid the above problems, and yet they have to be rather stupid in order to carry out such a mission, just so that we could see their exhaust polluting space around earth.
But aside from all that, if there were any high speed object traveling across your field of view, you would never be able to see it with the naked eye. At best, you would need extremely high speed photographic equipment to capture any details of the object. This is true even for an object traveling much less than the speed of light.