How would I go about replacing a CCFL laptop LCD scrren backlight with LEDs?
The CCFL inverter will most likely run off ~20V.  As long as you run your LEDs in such a way as to be able to handle that sort of voltage (multiple ones in series, series groups in parallel, current limiting resistor per series group), that should be fine.

As for the dimming?  The CCFL invertor may be being fed with a PWM signal to set the brightness of the CCFL (Some use PWM, some use an analog voltage - you'll have to probe around).  This PWM signal could be used to control the brightness of the LEDs by switching them off/on rapidly.  Then you will be controlling it in whatever way the laptop is designed to have its backlight brightness controlled.