Aktau is a city in the very west of Kazakhstan.
North towards Beyneu, Atyrau, Uzbekistan
The road between Aktau and Beyneu is rather bad, about half of it is not paved. Even with a good car, driving time can easily be 10 hours. It is very possible to get a direct lift to Beyneu or even Aktau though, as there is almost nothing in between. There are enough trucks and some cars all day. Hitching at night doesn't work so well.
If you overnight in the desert, be aware that traffic from Aktau leaves in the morning, so i.e. if you are 200 km from Aktau, there will be almost no vehicles before noon.
The road network around Aktau is spread out pretty wide, and there are many roads going to Shetpe, so it's kind of hard to know which road to take. The buses to Shetpe take the road to Shevchenko, passing the airport and then turning right towards Shetpe, into a road where Shetpe is not indicated on the signs. This is probably the fastest way and there is enough traffic, but maybe it's not easy to find it.
You can also take a bus to Shetpe. They go regularly from the main bus station in Aktau (go there by local bus 3) for 420 Tenge (ca. 2,10 €). From there it's quite easy to find a lift to Beyneu. Just walk to the end of the town, close to the train station, where the road leaves Shetpe.