The problem is likely not with your preamp itself (the circuit is basically OK), but with the power supply and gain setting.
Using the internal 5V from your PC (or USB port) for a mic pre is pretty much a no-no. There is almost certainly switching noise in that power rail which will be very audible. (The mic output is typically a few mV, perhaps 10-20mV or a bit more with an electret. This circuit has a gain of 100, so it looks like it expects about 50mV pk-pk as a maximum which seems believable.)
The simplest option is usually to derive a clean voltage rail from a linear regulator and use this for your pre. However if you start with +5V your rail can only be of the order of 4 or 4.5V - now you must select your opamp carefully so that it is suitable for low voltage operation, and also so that the output can swing as close to the rails as possible, so you do not lose headroom. (Pretty sure a 358 won't do this very well. The data sheet says that if you have +30V, Vout will go to 26V, at 5V they aren't admitting anything for the high side swing.)
A better approach is to use a switcher to derive one or more rails from the 5V (ideal would be about +/-16V) and then use linear regulators to get (say) +/-15V which you can use to power op amp circuits using standard opamps (like the venerable and still excellent NE5532). However this is obviously a more complex approach.
The second issue will be gain control. Just knowing the mic sensitivity is not enough. The difference between someone whispering and shouting into a mic can easily be 20-30dB, and on top of that distance of the source from the mic plays a big part (moving from 3 to 12 cm causes a 12dB change in signal). So a fixed gain mic amp is not much use. You need a gain control. (In fact to process a wide range of audio signals, some form of AGC or compression is usually used, but that is outside the scope of your question.)
Given these factors, your application is going to play a big part in your design decisions.