The solution below shows the connection line paths and certain components that holds for the LCD backlight LED to work., this may help in solving backlight white LED problem on Nokia 6303 classic after you confirmed that the LCD screen is okay or already tried replacing it.
Nokia 6303 LCD Display White LED Light connection Tracks
If you already confirmed that the LCD is working, or already tried replacing it but still it shows a dark screen display and the backlight LED is not working, you may then check the highlighted components above. You may start cleaning the LCD pin connector, re-solder it if necessary.
Here's some hints on how to troubleshoot the backlight LED problem:
Check the battery supply voltage(VBAT) across the inductor coil, check also the inductor coil for possible open circuit. Check/replace the 33 ohms resistor near the LED driver IC and the filter coil near the transistor.
Then check the LED+ and LED- voltages and trace each connection line paths from the pin connector down to the resistor and coil. If the voltage reading are missing, you may then try to replace the LED driver IC.