First you need a wav recorder, like Soundforge (or whatever you have will do), then you need a good MIC with no background noise. Or any MIC you have in most cases here lol.
Make sure your MIC is turned on if it has a switch and that it is also at a desirable level/not on mute in your sounds toolbar on the taskbar.
Record your soundtrack in clips: usually conversationally. So say character A says: Let's go to the park Eddy -- you record that and save it. Then character B say's: Sure thing Bub -- you record that and save it. You get the point.
After you have yoru entire soundtrack you can bleed them all into one wav if you are a sound producer (like me) and you can pan the voices and do the levels and mixing etc.. as you wish but this step is usually skipped as most people here on newgrounds are not sound people.
Once saved (or mastered) you must put them in your cartoon, I suggest streaming. You import the sound to stage, create a layer for your stream, and then in properties for your keyframe where it starts you select the sound, and make it 'streaming'. Then, you can animate the mouths to sync up with the soundtrack. This is the easiest method - to record the soundtrack FIRST then sync the lips, but if you already have the lips moving the process is a bit different:
First you should render your movie. Then, click record on your wave and well, talk away! It will probably take you a few tries, you may have record one wav for character A and another for character B, etc.. then sync them up in the timeline.
When you go to render your final flash, make sure the sound is set to render in STEREO this is a mistake EVERYONE makes on newgrounds because the default is MONO and mono sounds like SHIT.
Goodluck!