If you have a logic analyzer, connect it to all the inputs and outputs and you should be able to see it in operation.
Otherwise, your only option is to remove it from circuit and build a test rig for it. For simple logic functioning, use LEDs with 270 Ohm series resistors from each output to ground and tie all the inputs low with 470 Ohm resistors. If you then short each input combination to +5V you should see the corresponding binary output for the input you shorted.
If you want to do dynamic or parametric testing, you need a signal source and oscilloscope capable of resolving the rise and fall times of all the signals
and delays between them.
Brian.