How do I get from Changzhou to Shanghai airport?

There is no direct transportation. You can take a train (bullet one is advised) to Shanghai first, and then take a bus or subway to the airport. SH actually has two airport. Maybe you can tell which airport you are heading for, so I can tell you more detailed transportation methods.

I need to get to Pudong airport from Changzhou.

Then you'd better arrive at Shanghai Railway Station instead of SH Hongqiao Railway Station. It's closer to the airport.

Upon arrival, you can take subway line 1 to People's Square, and then change to line 2 to the airport. Or take airport shuttle bus line 5 to get there from the railway station directly.