It isn't possible to drive from the top to the bottom of Chile because the Patagonia ice field gets in the way, but there are still plenty of other amazing routes available. Driving through the northern part of the Atacama Desert is amazing at night, because the dryness of the air means this is the best place in the world for seeing the stars (in pictures of the Earth seen from space, the Atacama is the one brown patch amid all that blue and green). In the south of the country, you can explore ancient Araucanía, where the Mapuche were the last major Native American civilisation to hold out against the European conquest, and where they still dwell in isolated communities deep into the forest. There's a sense of timelessness in these places echoed by the coastal roads south of Santiago where you can look out across the ocean.
Chile is famous for having the best roads in South America (better than many in Europe) but there's a reason for this: many of them are toll roads. When you arrange your car rental, make sure you get a pre-pay device in your vehicle. This will give you the option of using these roads (it's not always possible to pay on site) and you can pay the charges when you return your vehicle.
Chilean drivers rarely give way and you may find that you need to drive quite aggressively if you want to change lanes, but otherwise they are courteous, making for a safe and pleasant driving experience.