FRAPS is the one I would use for a Let's Play, personally.
For those recorders that produce black screens, you could put the game in windowed mode (go into the graphics options) and record the window.
I know that Macs come with Quicktime, which actually allows you to record video very easily. (open it up, I believe it is in File > New Screen Recording or something along those lines)
Finally, for your microphone, you can run Audacity (a free open-source audio recording/editing program) while you record and then just add the recording as audio afterwards in a video editor. That way if you want, you can also mix in tunes underneath.
Best of luck. :)