Hi Detlef,
It's pretty easy. There are videos, books, and websites that can explain the process.
Here's a video:
http://www.buildersbooks.com/how_to_..._homebuilt.htm
If you are member of EAA (recommended) then they have a lot of help on their website.
Essentially you need to:
1) get an N number (from the FAA website, 10 USD)
2) register the airplane when you are about 6 months from the inspection (a couple of forms, very easy)
3) get a DAR or FAA dude to come out and inspect the airplane when it is finished.
It is *nothing* like what it is in Germany or some other European countries - much easier!
Here is the source of information on the FAA site:
http://www.faa.gov/licenses_certific...uilt_aircraft/
It looks much more complex than it is. Don't worry, when you get the US there will be lots of people who will be happy to walk you through the process.