I just picked up the 38SW-A serial adapter and software for the Amprobe 38XR-A -- which should be basically the same as your meter. I don't see an easy way to crack open the case, but I can say a little even treating it as a black (red) box.
First, it seems to communicate at 9600 baud, 8 bits, 1 stop bit, and no flow control or anything else. The meter talks to the computer but communication in the other direction isn't nescessary -- which makes things a little simpler. I think you only need to connect one of the lines. I haven't deciphered all the codes, but it doesn't look too bad. E.g. if the meter measures 78F, then it sends 0200
782A00080.
As for how to make it self powered, you can get power from the lines coming from your computer This product
datasheet talks a little bit about how isolation and port power works. You could probably use a switching capacitor inverter to get negative voltages from positive ones (or vice versa).
That said, although you could make an adapter to RS232, it's may be easier to do a USB adapter. That way, you'd get power very easily (from the usb) and you wouldn't have to deal with the negative voltages RS232 requires. FTDI is known for making USB to serial chips (I've heard they're fairly easy to use), but I'm sure others make them too.