How would I go about measuring an analog signal that has both positive and negative voltages on an arduino analog pin at a high sampling rate?
Using standard Arduino libraries, you will have trouble sampling above about 10000 samples per second.
If you manage to sample fast enough, you will need a place to put the captured data.  Memory is notoriously scarce on Arduinos, and the other built in option is the serial port to dump it to a PC.  You could also try to write it to an SD card.  Don't know what the libraries are like for that, or if there's a readily available SD card adapter for Arduinos.
The Arduinos have a 10 bit analog to digital converter.  Assuming a 5V reference, you are talking about 5mV steps.    For the range you describe (120mV) you would only be using 24 out of 1023 steps from the ADC.  You need an amplifier that multiplies by about 30, and that adds 2.5 V to the result.


If "less than 10kHz" fits your definition of "high sampling rate" and you can build a small circuit board for the amplifier, then you might be able to do something useful.