Remove the Hard Drive, and take a trip to your Local Recycling Place with the chassis of your old Lappy.
Put the Hard Drive into an Enclosure, available on eBay for a tenner or so and you will now have an External Hard Drive. Plug it into your new PC and copy the Files over.
Once you are happy that you have all your stuff safely stored on you new PC. You can format your new External Hard Drive and you can use it for Backups, which I gather you don't have.