How can I copy a ROM from a DS cartridge to my computer?

There is no way to dump DS cartridges without using either a DS and some kind of flash cart or perhaps a 3DS and homebrew... unless you are willing to make your own custom hardware (no simple task). The closest thing might be the NDS Adapter Plus, but that supports only reading and writing save files, not the games themselves.

However, it has been a few years. Perhaps you could manage to get a DS these days -- they are much cheaper now. If you do that, then you will still need a flash cart of some kind to run a custom program to copy the cartridge, not to mention the cartridge itself. This gbatemp forum thread has instructions for various dumping setups.

Alternatively, you can use a homebrew-capable 3DS with the above-linked software, too.

In my case, it was an M3 Perfect slot2 flash cart using wood dumper (slot2). Run the dumping application from your flash card, insert your new Pokemon Black cartridge, push the button, and with luck you will have a copy of your game. (I don't think this copies save files, though, so if you want to do that too you'll have to find another tool for that -- but there is hardware to do it.)

You can then copy the dumped .NDS file and play it in your emulator of choice. You might need to clean up the dumped ROM, though, as in my experience it tends to copy too much data (effectively, garbage).

There are also other options, such as using Wi-Fi to copy a ROM over the network.

Of course, if you did all of this, then you already have the game you want to play and a Nintendo DS... But nothing stops you from playing the copy in an emulator.