Mount the old drive if it is bolted into the box with the new one (two hard drives in the same machine) and use the file browser to copy-n-paste
A cleaner method, one I've used more than once. Use tty Linux
. Burn the ISO, bolt the old drive into the/a machine, boot from the CD, log in (instructions are on the screen) and mount the hard drive.
to see what the drive is called
to see where you can mount, and mount the drive
mount /the/device-found-by-fdisk /mnt/c:
and CD to the folder, have a look-see if everything is still there...
Give the box a USB stick, mount the same as the drive (fdisk - mount)
copy from one mount point to the other