You need to get the entire iTunes folder including the library files (e.g., .itdb, .itml, .xml) onto your new computer. So, you can:
1). Copy the entirety of the iTunes folder from your old computer onto an external hard drive, USB stick, etc. Then, you guessed it, copy that from your temporary storage space onto the new computer. Ensure you are signed in to your (old) Apple ID account, and that the new computer is authorized to play purchases.
2). If you don't have somewhere to dump the folder to, you can use Migration Assistant on PC or Mac.
3). iTunes Match ($$$; beaten by allegro).