Huh? The bias of that node is already established by R5.
"A lot of noise that goes away when I connect the ground wire" is likely caused by either
- a very large ground loop with the wire not in place
- no common ground between the connected parts, so that the ground is only established by parasitic capacitive coupling.
~> Did you connect the ground of the cinch or TRS plug (or what you used to connect the output) to the circuits ground?
Also, as Hero999 already said, there are many op amps far better than the 358/324 for audio. The 5532 would be a classic and is both cheap and well suited for this kind of amplifier.
A FET-input op amp doesn't improve things here over a bipolar one, since the input impedance is low (1 k?) - whether this is bad or causes significant attenuation of the signal depends on the microphone, but with common types you are looking at an attenuation of maybe 2 to 4 fold ; thus the noise gain - signal gain ratio of this circuit is poor (101 * attenuation / 100). I suggest making the circuit non-inverting, biasing the (+) to C1 node explicitly.
There should be more than plenty of schematics to draw inspiration from, but make sure to filter audiophool stuff out... ESP probably has something well-thought out.