If you have access to an SMU or a calibrator it’s fairly easy to write SCPI to go through the entire input range and calculate gain and offset error and integral nonlinearity. If you do not have a programmable source you can build one with a DAC, even a 10 bit one suffice, and measure the data points with a SCPI capable multimeter.
Another, perhaps the first step on the plan is a zero/or mid FS measurement, to record a histogram of zero reads. It would give you a picture of noise, either from external sources, like reference or power, or internal quantization, etc. of your device. You’d like to get this close to the data sheet specs, to make sure your test setup is acceptable.
If it’s a fast ADC you can characterize linearity, missing codes, etc. by taking a histogram of a low distortion sine wave.