I did not need to rectify my power source. People S 250 uses DC for headlights. In your case, you will need to use diodes or bridge rectifier to turn AC into DC. Since the relay coil does not draw much current, pretty much any rectifiers will do. The HID will use the DC current from the battery so the wire from the relay switch to the battery should not be too thin.
Do you have a multimeter? If you have not used a multimeter before, I suggest you get someone who are familiar with basic electronics to help you. I may be able to draw you a wiring diagram if you want.