Two methods, both require a program called Fraps.
The first is turning off your HUD/Radar, getting a suitable vehicle, then press record and do some stunts.
The second is by editing your text file (x.gxt where x can be american, spanish, french, etc....) so the word REPLAY isn't shown during replays. (More on that here)
You then do a suitable stunt and press F1, then press record.
The first method gives you more camera control.
The second saves time, gives better/more footage in less time, and is generally preferred.