The amplifier is horrible but it will work if the transistors are carefully selected, what it doesn't do is cater for variations in transistor parameters and as Audioguru pointed out, it's input and output characteristics are far from ideal.
Your basic problem is that an ADC can't measure negative voltages, meaning any voltage below it's ground pin. If you just remove the DC and leave the AC part behind, half of it will be above and half below ground voltage so at best you will see a rectified measurement (= very high distortion). There are two things you have to do, the first is remove the earphone from the circuit and replace it with a fixed resistor having the same DC resistance. If you leave the earphone connected it will behave like a second microphone and may even cause acoustic feedback, if you simply remove it, the amplifier will no longer work. The second thing is to couple only the audio waveform to the ADC and you correctly assumed a capacitor would do that - however, that brings us back to the +/- problem again. To fix that, you have to apply a fixed offset to the signal, basically 'lifting up' the whole waveform so both it's positive and negative peaks are within the range the ADC can measure. The easiest way to do this is with a potential divider across the ADC reference voltage and ground. If you are using VDD as the high ADC reference and ground as the low reference, all you need is two equal value resistors across the supply pins. You connect the capacitor to the junction of the two resistors and the ADC input. Under no-signal conditions, the ADC should read half FSD, for example if you are using a 5V supply it should read back 2.5V. As the audio adds to the half FSD, it should make the ADC reading increase or decrease according to the audio waveform and that's exactly what you want!
Your next problem will sampling the ADC fast enough but lets cross that bridge when we reach it.
Brian.