For pulse width measurments you can start looking at NI examples located in Search Examples >> I/O interfaces >> DAQ >> Counters >> Pulse width measurments. The 6024E is equipped with DAQ-STC counters, and there are two examples to look at.
I can't figure out how to measure the rising time of a square signal with a counter. Maybe you can acquire the signal in an analog channel at the maximum scanning speed and measure its rise time by searching in the waveform obtained...
You can try posting this question to the Counter/Timer section of the Developer Exchange.
Roberto