Use a digital multimeter to do this. Set it to the 10A setting and disconnect the inverters positive lead from the battery. Connect the positive lead of the multimeter to the battery and the multimeters negative lead to the inverters positive lead. The display of the multimeter will now show the current draw.
Charge controllers use very little power. Mine uses 10mA (0.01A). I barely reads on my meter. However to find out, use the same method as above for the inverter.
You should always turn off or disconnect the inverter when its not in use because they draw a stupid amount of idle current. Mine uses around 1A when idling and my system simply wouldn't be able to cope with that sort of draw 24/7. Remember that renewable energy is also about conserving power. If you don't have the power, don't use it, and disconnect anything with a parasitic load when you're not using it (eg the inverter).
Overnight my inverter will waste 10-12Ah of power. My solar only generates that amount per day, IF I'm lucky. That 10-12Ah of power could be used to run my desk light, or my phone charger, or even my laptop for a few hours. Inverter idle currents are very wasteful.