Make sure your iPhone is fully charged before you try to initiate the transfer.
After checking the battery is sufficient enough, plug in your iPhone to your computer.
Now you should open iTunes. For MacBook users, double click the iTunes icon on your desktop and PC users should click on ‘Start’ and choose ‘iTunes’ under the ‘All Programs’ category.
In the iTunes panel, select your device name to access your iPhone on your computer. Now you need to click on ‘Restore’ in order to hit the ‘Backup’ tab to download your backup to new location.
After the backup is complete, you can go and check the size of backup file at your new location to have an estimate whether the backup has been done properly. Usually, no problem occurs in this task but it is always good to be on the safer side.
In order to move your old backup from your previous computer you can use a USB device or insert a new writeable CD or DVD into your DVD-ROM. Drag your backup files to your new CD and click the ‘Burn’ button to initiate the writing process.
After the process is complete, take out your disc and insert into the new computer and Copy the files to whatever location you wish to save to.