Probably the best way is to copy your iTunes back up from your original computer to your new computer.
Here's a step by step guide on how to do that:
http://osxdaily.com/2012/02/29/sync-iphone-new-computer-without-losing-data/
"Syncing an iPhone with a New Mac by Copying Sync Data
Quit iTunes on both Macs and disconnect the iPhone from both Macs
Open the Home folder and copy the iTunes directory from the old computer to the new one, located at:
~/Music/iTunes
Now open the user library directory and copy the backups from the old computer to the new one, located at:
~/Library/Application Support/MobileSync/
To be certain everything is in order, copy over iTunes Preferences files as well:
~/Library/Preferences/com.apple.iTunes.plist
Now connect the iPhone to the new Mac, launch iTunes, confirm that everything is in order, and authorize the new computer"
Disclaimer: I've never actually tried it that way, but it should work.
Also, I think if you plug your iPhone into the new computer you are given the option to back it up to the new computer. Again, I haven't tried that one but it's worth a shot.