Knowing which model Mac the old one was would help as the type of hard drive varied by model.
If it was the CPU, or motherboard, (it won't power on at all) the hard drive is probably still good.
It is a simple matter to remove the hard drive and buy an external case for it. (It looks scary, but anyone with a couple of screwdrivers can do it.) Removing a hard drive is covered online, step by step for the different Mac models. A search for "hard drive removal" (and add your Mac model, like iMac, Mini, PPC 8600, etc.) will find numerous links. Since the Mac is already dead, you won't hurt it by opening the case.
The cases are available in USB and Firewire external connections and a number of internal connections. If you bring your old hard drive into a computer store, they can look at the connections and tell which case you need. (and probably put it in the external case for you.) I've seen USB cases for under $20 and Firewire for under $40. Either would do what you need, so I'd go for the lower priced USB.
http://eshop.macsales.com/item/EZ%20Quest/N33300/ shows one example of a USB case for the hard drives used in most of iMacs, Minis and Towers for the last 4 or 5 years for $19. If you call them, they can tell you which case you need by your old Mac model, if it is bootable and probably give you a link to how to remove the drive and put it in the case.
If you buy a case that is "bootable" it can even be used to start your MacBook as a back up to your built in hard drive. I do that for maintenance of the built in hard drive and even to run a second OS. For instance, you could have OS 10.6 (Snow Leopard) running Windows via Bootstrap on the external drive and not fill the built in drive with all of the extra Windows stuff.
Then simply plug the USB or Firewire cable from the case into your MacBook, plug the case's power cord into a wall outlet and it will act as a second hard drive. From there you can copy the iTunes library and all associated files to the MacBook. After copying everything you want, you can reformat the drive and have an external drive for storage, etc.