Connect the iPhone to your computer, launch iTunes. From the menu bar click File / Transfer Purchases From (your name) iPhone.
If you need more help.. iOS: How to transfer or sync content to your computer
You can only transfer items purchase on the phone to a computer. The iPhone is not a backup device.
Your best bet is use the backup of your old computer to move your iTunes library to the new machine, along with things like your contacts and calendar if you're syncing them locally.
Once this is done, you will need to set up the phone to sync with the new machine. This will erase any media on the phone, but it should not be a problem as the media will already exist on the new computer.