It is often possible to sleep overnight in big airports. Airports are safe and warm places, in addition you can always find there free toilets where you can wash yourself. Sleeping is generally tolerated by the security staff since many persons have early-flight departures and have nearly no other choice than sleeping there.
As a hitchhiker, you can notice the airports on a map and leave the car when it goes near it. If you want to visit a city, it is often the cheapest place where you can sleep (I say cheap, not free, because you will often have to pay for the transport to go there.)
It's a good advice to have a mattress, a sleeping bag, earplugs and something to put on your eyes as there may be loud announcements and light during the whole night.
If you want to know in which airports you can sleep, check the website SleepingInAirports.com.
Airports are often close to big roads or highways. You may try to catch a ride from someone leaving a parking or a taxi going back to the city after dropping someone at the airport.
Another possibility is to hitchhike in the arrival hall. Many people waiting for other people stands with signs like "Mr Robinson". You can do the same, but instead of writing a name, write your destination: "City centre", "South"... People freshly landed or people waiting for someone will be quite receptive to such a sign and may approach you and offer you a ride.