@coincheung
Check out https://en.wikipedia.org/wiki/Kullback–Leibler_divergence#Properties 68
The important point to note here is P and Q are probability distributions, so even though the value for a particular point in the sample space (discrete-case) is negative, the summation over all points in the sample space must be non-negative.
Look at https://en.wikipedia.org/wiki/Kullback–Leibler_divergence#Basic_example 100 for an example calculation.