How do I verify Philips curve empirically for recent years? I wanted to verify the relationship between inflation and unemployment empirically, so I took data from the World Bank and used excel to correlate them. 

I got a very, very bad correlation between unemployment and inflation (from 1991 to 2016). Am I doing something wrong here?

I took these two sets of indicators: (1) Unemployment, total (modeled ILO estimate) and (2) Inflation, consumer prices. Here are the links: http://data.worldbank.org/indicator/SL.UEM.TOTL.ZS and http://data.worldbank.org/indicator/FP.CPI.TOTL.ZG

I downloaded the data from here and I studied the data for the United States of America. Is the low correlation because of the fact that the market needs time to react to unemployment rates? If so how did Phillips study it in the first place?
Simple correlation is not a good approach to evaluate the Phillips curve, because you are mixing many periods together. To get a better grasp of what this relationship might have been in the past, use a graphical approach. 

I suggest you to do something like this:



So basically, you plot the points, but add a line that connects them temporally. This might give you a better idea of some dynamics. It is clear from that graph that a simple correlation cannot capture the complex dynamics of inflation and unemployment.

For example, this is an animation of the evolution of the Phillip curve for the US:



The author identifies three or four PC over 40 years, reflecting different political/economic periods. A simple correlation analysis however would show none.

Source of images: here.