What do I need for my arduino + current sensor to do data logging ?

Is this a high or low voltage application?

Do the test device (Arduino) and the system you are testing share the same ground?

Where do you want to log to, and how?

To resolve 0.1 mA and 1 Amp with the same sensor, you need a 16 bit A/D, which is beyond a normal Arduino. There are current sense chips that have a high-resolution built-in A/D converter that can interface to some Arduinos. Such as a TI INA237.

And not just that.  You need to check with the datasheet about the error rates because some can be quite high and all will have some impact on readings at 100uA.  I think you may need to do some form of standard deviation processing on the data presentation to better qualify what the results are showing and take account of outliers.  You mention 1 to 5 seconds of data but at what resolution, e.g. once per 0.1 seconds?  Check the sample rate at 16-bit vs your timing resolution vs accuracy.  

Ultimately, how you log will depend upon the volume of data points that need to be logged, how real-time the trend needs to be observed etc.  I think more info is required to help more.

As other members have mentioned, it is better if you use current sense ICs. They will give you better resolution as well as the correct reading.
To store/log the data, you can interface a SD card with the Arduino. If you need the data along with the timestamp, you might consider interfacing a real-time clock(RTC).

That's non-trivial, and as Andrew mentions, there is the issue of noise that could swamp your lower readings, and you can try to remove some of that through filtering by say averaging, but then the speed suffers. 

There's an example project here https://www.youtube.com/watch?v=dBKbuUVFMk4

which used an 18-bit ADC, for the same purpose (logging at low and high current) as part of a larger project that also includes voltage sourcing, but if you examine the circuit on the project page, the current monitoring portion was almost half of the entire project. That project may meet your needs (by omitting the voltage sourcing portion of the project), however most of the parts for that project are likely unobtainable currently due to the semiconductor shortage.

Here is how averaging can make a difference. You can see from the video, that the averaging can show a nice flat line at very low current, but the instantaneous measurement will have noise. The x-axis is in seconds:

Here you can see what I think were individual measurements, scattered on the display, so you can see the actual noise on the measurements; I don't recall what the x-axis period was:

Sorry to do this, but I’m commenting to test: suddenly I have to become a member of Arduino to participate, despite posting earlier.

Weird. Registered Users should be able to reply without having to join first.