You'll need a DAC AND a 3.5mm to stereo phono RCA cable to get the most of your BHC.
Alternatively you can just get that cable, hook it from your onboard sound onto your BHC and see if you like it. Computers have "clean" power in terms of getting power to electronics, but are riddled with tones of oscillation which can be easily picked up by ears alone. Speakers are powered by electricity, you can imagine how it'll sound if they are powered directly via the computer.
Now apply this to an amp, which by default should just amplify whatever it's been fed. If it is full of noise, then you'll get louder noise in return.
Most USB DACs are powered by USB, but also include their own power regulator that removes a significant amount of noise, hence why a good amount of us like external DACs over PC sound cards.
A DAC I began and could recommend for the budget minded is the ELE D01, which performed admirably. That should provide a low enough noise level to prevent things sounding like it's constantly raining, without breaking the bank.