we agreed on a sampling rate of 1Msps. 200ksps is the absolute lower bound, however.
The little detail that changes everything...
You can't do that by wiring an ADC to your computer directly. The computer uses a multitasking OS which is unable to interrupt 1 million times per second to trigger a conversion.
You need some hardware interfaced with your ADC, which will run it at 1 Msps, buffer the data, and forward it to the computer via a connection of suitable bandwidth like USB or Ethernet.
The easiest would be to use a digital storage scope with a record function, or a streaming USB scope, or a data acquisition (DAQ) system.
You can use a USB microcontroller too, provided it has enough USB bandwidth and power, and the proper interface for your ADC. This will be quite a bit of work though, but you can probably find an ADV eval board with all this stuff. This also needs software on the PC though.
If your computer has audio inputs, you can also use these as ADCs, but you will need to remove the AC coupling caps, and perhaps the software highpass filter, that depends on the chipset. That won't give 1Msps though.
Another solution would be to lowpass the signal to reduce the sample rate, but you'll lose time resolution.
Before designing (or buying) an acquisition system, it would be wise to check what the signal actually looks like. Especially its bandwidth. Therefore, try a digital scope first.
Now, about bandwidth... AD8211 has a pretty good bandwidth (300kHz) so if you use a low value shunt resistor you will be able to exploit the full bandwidth. However the resistor will be in series with the supply, which will cause the voltage to drop a little bit.
This should not be a problem as your SSD won't be powered by the +5/+12 lines directly, rather it will have a few switching regulators onboard to generate lower voltages. So if the supply drops by a fraction of a volt due to the sense resistor, it will still work fine. The current measurement will be a bit higher than without the sense resistor though, as a switching regulator powered from a lower voltage will draw more current at constant output power. So try to use a small resistor value.
If you include an additional regulator as suggested by Toni, then your measurement bandwidth will be that of the regulator (ie, a few kHz unless you take some desperate measures).
Whether you actually need this huge amount of bandwidth remains to be seen!...