You should be fine. Just trasnfer the disk to the new computer.
If you need to reinstall for any reason, the data in home is preserved as long as you do not format the partition with /home on it.
See https://help.ubuntu.com/community/UbuntuReinstallation
Since Hardy it is possible to reinstall Ubuntu without losing the
content of the /home folder (the folder that contains program
settings, internet bookmarks, emails and all your documents, music,
videos and other user files). This can be done even if /home is not on
a separate partition (which is the case by default if you did not
manually separate it when installing Ubuntu originally). This tutorial
can also be used to upgrade Ubuntu (eg 11.04 -> 12.04 from a 12.04
live-CD).
Of course if you value your data it should be backed up regardless
See - https://help.ubuntu.com/community/BackupYourSystem
All your user data is in /home so you can copy or rsync /home to a DVD, flash drive, etc from a live CD. Be sure to use a Linux filesystem to preserve permissions (DO NOT BACK UP TO VFAT OR NTFS).