Measuring the noise voltage requires two basic objects:
1. A filter to define the bandwidth. Ignoring hum for the time being, a simple R-C low-pass filter suffices. Note that the equivalent noise bandwidth for a single-pole filter is 1.57 x (bandwidth at -3dB). see
https://www.k-state.edu/edl/docs/pubs/technical-resources/Technote1.pdf2. An AC voltmeter, preferably true-rms. If you use an average-responding, rms-calibrated meter, there is a further correction factor required, since the rms-calibration assumes a sine wave. This is about 0.89: see
http://hpmemoryproject.org/an/pdf/an_60.pdf for a classic treatment of the different styles of voltmeter and their response to non-sinusoidal waveforms (noise on page 6).
A basic treatment can be found at
https://pdfs.semanticscholar.org/d770/d93b92abf067aab6e469ba816da0d205777e.pdfWeighting filters are normally used with acoustic noise measurements, to simulate the response of the human ear. For amplifier signal-to-noise ratio, you measure the noise in the relevant bandwidth (probably 20 kHz for an audio amplifier) and compare it to the maximum amplifier output or to the typical amplifier output, as required.