There are several ways to measure bias: measuring the bias voltage, using an oscilloscope, and diretctly or indirectly measuring current.
Measuring the bias voltage seems to be the least reliable because it doesn't address any of the variables in the tubes. It only applies to fixed biased amps, and it involves measuring the negative voltage being applied to the grids. I've had tubes that were biased correctly, yet if you were to stick in another set of tubes they would die at that setting. Measuring the bias voltage wouldn't have told you this, the other methods would.
Using an oscilloscope requires some test equipment--the scope, a signal generator and a dummy load. I haven't used this method, but you run a signal into the amp while it's hooked to the dummy load and watch the output. On push-pull amps, you'll see a flat spot between the positive and negative halves of the wave. You adjust the bias to eliminate this notch.
What I've read is that determining when the notch is gone is somewhat subjective, and generally leads to a bias setting that's rather cold. Also some types of tubes have more rounded edges on the curves which adds to the difficulty. I don't know how this method works for single-ended amps.
Measuring the current drawn by the tubes is the third way. Here's my experience with 2 methods, transformer shunt and cathode resistor.
Transformer shunt
With transformer shunt, you hook your meter so that it shorts half of the output transformer. This involves putting the red lead of the meter on the center tap of the output transformer, and the black lead on the plate. The center tap usually connects to a large filter capacitor and/or the standby switch, but you'll need a schematic to be sure where to find it. The plate on octal power tubes is pin 3 for the common ones.
Since the meter should have lower impedance than the transformer, nearly all the current flowing through the tubes should flow through the meter instead. The meter then measures the current flow. As long as the meter really is lower impedance, this works. On amps with 2 power tubes, you're reading the current through one of the tubes. On amps with 4 power tubes, you're reading the current through 2 tubes. Therefore the reading you get will be the sum of the current drawn by these 2 tubes. If the tubes are balanced, you can divide this by 2 to get the bias current.
A caution: the plate voltage of several hundred volts is present at the meter, so use alligator clips to wire the meter with the power off. Don't trust your ability to hold the probe on the test points. It's too easy to slip and short the plate voltage to ground or through yourself. Also, since you've shorted half the output transformer, you can't play through the amp without unhooking the meter. This makes it difficult to compare bias settings, since you have to power off the amp and disconnect the meter before powering it up and playing.
To get around this, you can correlate the bias current to the bias voltage. Set the bias (using the transformer shunt method) so the current drawn by the tubes is in the low part of the accepted range. Unhook the meter, turn the amp back on and measure the bias voltage that develops this current. Now re-bias to the upper range using the transformer shunt method. Unhook the meter, power up, and measure the bias voltage again. You now have 2 voltages that correlate to the upper and lower current draws. Using alligator clips, hook the meter up to read the bias voltage. You can now play through the amp while tweaking the bias. Just keep the voltage being read by the meter in between the 2 readings you just obtained, and adjust the bias to a sound you like. Once you've zeroed in on a sound, I'd check the bias current again just to be sure.
The problem I had was with a Marshall Super Lead. The transformer shunt method gave me current readings that were lower than they actually were, and I was setting the bias so the tubes were drawing a lot of current. How did I discover this? By using the cathode resistor method. (By the way, I tried 2 meters, both read incorrectly, and both were accurate when used on Fender amps. My best guess is that some current is flowing through the output transformer, so the meter only shunts a portion of it and therefore reads low.)
Cathode resistor
This method inserts a 1 ohm or 10 ohm resistor in between the cathode and ground and uses Ohm's Law. The current through the cathode is the same as through the plate. When this current flows through this small resistor, a voltage is developed. So if 38 milliamps is flowing through a 1 ohm resistor, 0.038 volts is developed. In cathode biased amps, this resistor is already present, and is usually 250-500 ohms. With fixed bias amps, this resistor can be added as a permanent modification to all the cathodes, or can be part of a bias probe. (Plans for a bias probe are available on R.G. Keen's Web site, or bias probes can be purchased). 1 ohm and 10 ohm resistors are nice because you don't have to do any arithmetic, but any low value could be used.
I now use this method because:
it's a low voltage measurement
you can monitor it while playing the amp
you can measure the current on any tube in a 4 tube amp and check on how well they're matched
I can bias a Marshall Super Lead