You would use Conditional Formatting for this. Suppose your Due dates are in column D, starting with cell D2, and the completion dates are in column E, also starting with E2 (to allow a row for headers). Then you should select all the cells from E2 to the bottom of your list, then click on Conditional Formatting | New Rule | Use a formula... , then put this formula in the dialogue box that pops up:

=$E2>$D2

Click on the Format button | Fill tab and choose the colour you want to use, then click OK twice to exit the dialogue box - Excel will adjust those cell references automatically to suit the cells that had been selected, so you only need to apply this once.

Hope this helps.

Pete