What is the proper way to ground this DC circuit?
In electronics, Common and Ground are the same thing and they just mean 'Reference 0V'. This is because, in an electronic circuit, most voltage measurements are made with respect to a common point in the circuit. The common point could be Vcc too, but it is more convenient to use the -ve terminal of the Power supply and call it GND or COM and make all measurements with respect to it. Do not confuse GND in electronics with 'Grounding' or 'Earthing', a thing done with electrical appliances to protect the user from a shock in the even of a fault.
You can simply connect all the GND and COM pins of various ICs together and to the -ve (0V, black terminal) of the PSU and you will be fine.
This terminology often confuses beginners. I too had pretty much the same doubt when I started building electronic circuits.