While a car can't go the speed of light, we'll treat this as a thought experiment in special relativity.
Let's imagine the person in the car has a meter stick pointing in the direction of travel, and is also holding a clock.
The person in the car would see the light streaming away from the car at the speed of light. (This is because light always travels the speed of light in any inertial reference frame.) That person would also see a stick that is one meter long, and will see the clock ticking normally.
An external observer that has the car passing by at the speed of light would see the following. The light from the headlights is traveling at the speed of light, even though it's the same light seen in a different reference frame. However, the car is also traveling at the speed of light, so the observer "sees" light that never leaves the headlights. Meanwhile, he observer sees a meter stick that has zero length, and the clock is at a standstill. This is a result of length contraction and time dilation, in other words, space and time adjusting to keep the light traveling at the same speed, even when it is seen from different reference frames.
And here's an example if the car is moving at half the speed of light.
Again, the person in the car sees the same as above.
Now, the observer sees the car passing at half the speed of light, a meter stick that is 0.866 meters in length, and a clock running at one tick every 1.155 seconds.
The conversions come from the application of the Lorentz factor, which is sqrt(1 - (v^2/c^2)).
[I don't get why people offer an answer if they don't understand this classic thought experiment in special relativity.]