There are several ways to test an AC capacitor. One of the best is to use a multimeter. Set the meter to a capacitance setting and connect the red probe to the positive terminal of the capacitor, and the black probe to the negative terminal.
If the meter reads close to the rating label on the capacitor, it is probably in good condition. However, if the reading is off by more than 10% of the capacitor's rated capacity, it is likely a problem.
To check the capacitor for continuity, you can use a battery-powered multimeter. Set the meter to the high ohms range (somewhere above 10k and 1m ohms). Touch the meter's probes to the capacitor's leads, both red to positive and black to negative.