Electret microphone amplifier for computer using transistor
The first and most obvious thing is that electret microphones need a voltage source to operate.

You need to put in a resistor from 5V to the point you have marked "sound.". 

The recommended value can usually be found in the datasheet.  If you don't have a datasheet, then a value of 1kOhm to 2kOhm should be close enough to get you started.

The rest of the circuit doesn't look really all that good, but should work.

I assume you have connected the cable for the PC in place of the speaker.



There is a way to power the amplifier directly from the microphone input on the PC.

PC sound cards always have a DC voltage present on the microphone input.  This is because electret microphones require this voltage to be present.

You need a relatively small resistor (a few hundred Ohms) and a relatively large capacitor (10microfarads or so) to make a low pass filter connected to the PC.  This gives you a (somewhat clean) DC that you use to power the amplifier.  The output of the amplifier then goes directly back to the PC (amp out goes to before the low pass.)

For voice recordings, this can work well.   

You need to make sure that the low pass for the DC is below the high pass in the amplifier (cutoff frequency wise, that is.  They don't care much where they are physically located.)

A quick look around google didn't turn up an example, and since I'm posting from my phone I can't draw one for you.

It isn't complicated, but words make things sound complicated when diagram would show you how simple it really is.