If the games haven't already been dumped, get ready for some TINY wiring!  Inside the clone will be a NOAC (Nintendo on a chip) in a blob, and the ROMs/mapper in another blob.  You will have to physically cut out the NOAC, then wire the ROM to an existing NES cart.  It will look something like http://sealiecomputing.com/images/intvnoac2.jpg when you are done.  The green board is from an Intellivision plug and play system, and the red is my ReproPak board.  If you are really lucky there will be a separate ROM chip instead of just a blob which makes rewiring easier.

Once you have done that you can connect it to a CopyNES for reverse engineering and dumping.  Depending on how the mapper works you may be able to separate out the individual games and put them on a standard donor board.

The wiring is connecting the plug and play system to a standard NES cart board, so your new board just plugs right into a NES and plays from there.  You need the CopyNES part to dump the ROMs to your computer to separate the games. 

If the system uses a real ROM chip and a separate mapper chip instead of just one blob you should be able to wire it to a normal chip programmer instead of using a CopyNES.  You still may need the CopyNES to reverse engineer the mapper tho.

The problem is you need to figure out what mapper to use, which can only really be done by watching the code run.  Can't watch it run without dumping first, which is why you need the CopyNES.