Is it possible to check the value of the voltage (hence the current) with such a high-frequency waveform? How fast can the ADC in your microprocessor provide an accurate representation of an analog signal?
Assuming the scope shot you've provided shows a 10 µs/div scaling, our ADC's can keep up with this signal. We recommend using an ADC with at least a 1 Mbps sampling rate. You can use our MAPS tool to narrow down the 16-bit device which meets your needs.
Will the low value of the voltage across the sense resistor be a problem?
Assuming 1.72 A on the scope shot is the amplitude of the waveform, this shouldn't be a problem for the controller ADC.
If you believe that this is achievable, is there a recommended sensing circuitry?
If 1.72 V is the peak voltage seen across Rsense, you may need to pick a resistor which is capable of dissipating the power you are putting through it. Using P = (V2)/R = [(1.72)*(1.72)] / (0.36) = 8.2 W. Currently, you have a 2 W resistor for Rsense. Also, we recommend that you put a buffer on the CS node (input to the micro ADC) to keep the holding cap on the ADC channel from loading the signal.