There can be no sensible answer, because it's impossibly in the first place under current theory.
However, if your car was travelling at a speed very, very close to the speed of light, but not quite, then the light from your headlamps would move away from you at the speed of light.
For matter to reach C (the velocity of light) is theoretically impossible. The energy requirements of obtaining C exceed that of the "Big Bang". At an infinite rate of acceleration it would take an eternity to reach the velocity of light. Time dilates and distance contracts to zero at the velocity of light.
Clanad's link is an excellent illustration of the problem of obtaining C.
To properly understand relativity consider what happens at velocities approaching C. If from a point of rest you accelerate to 90% of C you have not even begun to make any progress in obtaining C. At any velocity you acquire from a point of �rest� time and distance are correspondingly altered so that C remains C from your new position. In affect, your new position becomes a new point of rest relative to C from which you must again accelerate. As observed and measured from your original starting point you would have a velocity of 90% then 99% then 99.9% then 99.99% etc. never actually arriving at the allusive velocity of C (This is a simplification but hopefully gives some insight into the actual problem of accelerating to C.
Matty 443 was not asking whether it would be feasible to reach the speed of light - he was advancing a 'what if?' question.
The answer to the query as posed is; 'you wouldn't notice any difference from the situation with which you are familiar'.
It's a matter of relativity. Relative to your car, the beam of light would appear normal, but relative to circumstances elsewhere, both the motion of the car and its headlight would appear to 'freeze' in time.