With a simple multimeter, you can do a basic static test of the capacitor. All this tells you is if the capacitor is shorted or not. Basically tells you if it's bad but not necessarily if it's good. You should first short the capacitor out with a screwdriver to make sure it doesn't have a charge on it that could damage your meter. Set the meter to read ohms and place the probes across the terminals. Initially, it will be a short circuit and read zero or very low ohms but it will quickly charge up so the reading should be increasing in ohms value. It's best to do this with an analog meter so that you can see the meter needle swing as it charges up rather than seeing a bunch of increasing numbers on the digital meter.
To get a better indication of the condition of the capacitor, many multimeters have a capacitor check function that will read the value of the capacitor. Something that just an ohmmeter cannot do. Here, I am showing this type of meter reading a capacitor from a GE machine. It is showing 47.39 microfarads which is within tolerance of the 45 mf capacitor.
The problem with these static tests is that you are only using the low DC voltage of the meter to test the capacitor while in actual use, they will have 120vac across them. I have a piece of test equipment that will do full dynamic testing of capacitors that place full rated voltage across them and check for value, leakage, and ESR (resistance). This can find problems that a multimeter cannot. Most tech's don't have this type of equipment and really don't need it. If it doesn't look like it's been cooking, doesn't smell bad, isn't shorted (seems to charge ok with multimeter), it's probably ok. Best to just carry a couple spares to do a quick swap check anyway. You cannot bypass the capacitor as the motor needs it to start. You can however, disconnect the capacitor, apply power to the motor and quickly start it by hand to see if it functions. On Whirlpool motors, the capacitor is switched out of circuit anyway as soon as it gets going.
As far as the motor windings, overload and switch, you can do some basic ohmmeter tests. For the Whirlpool motors, you should read 4 to 7 ohms across the start winding (yellow and black wires), 3/4 to 2 ohms across the high speed windings (blue and white wires), 1 1/2 to 3 ohms across the low speed winding (white/violet and white wires), and 1 1/2 to 3 ohms across the extra low speed winding (white/orange and white wires). You can check the overload switch between the white/black and white wires which should read dead short (zero ohms). With the motor switch in place, you should read short (zero ohms) across the red terminal and black wire (start winding switch) as well as the same across the orange terminal and blue wire. You must remove the switch to further test the switch mechanism. With the switch removed, you should have open circuit between red terminal and black wire and open between orange terminal and blue wire and dead short between orange terminal and violet/white wire.
Power applied to a motor that won't rotate can be bad on the motor windings and capacitor. The capacitor is only meant to be in circuit for just a second or two til the motor gets up to speed. It is then switched out of circuit by the motor switch. If the motor won't rotate or the switch fails, the capacitor can quickly be destroyed.