The old computer is still viable? Lightroom does not really like it when you do things outside of it's own software.
You may need to install LR3 and than upgrade it to LR4 and finally to LR5 on the new computer.
I would also copy each catalog (backups) to the new version(s) and computer.. Maybe you only need LR4's stuff. But since LR3 was your only full program, I still think I would go through the process.
Of course you changed computers but it is always best to use LR to do any moves or copying and not any file manager outside LR.
Actually, you don't have to re-install and upgrade. You can just install the latest version of Lightroom. Then open your old catalog using the new version. Lightroom will ask if you want to upgrade the catalog. Choose YES, and it will do the rest. Also, when upgrading, it will make duplicate the old catalog so in any case if the upgrade is failed, you still have your old untouch catalog. There is no point of installing many version.
Back to ScottyP, all lightroom edits are stored in its catalog. So you would have to copy the catalog on your old computer to the new one and open it with Lightroom 5. Then, it will upgrade your catalog to LR5 and you can open it.
Since you store all your photos on an external drive, you may need to change the drive letter of the drive on the new computer to match to old one; otherwise, LR may say that your photos are missing. For example, when you plug in your external HDD into the old computer, the computer assigned it drive letter D. Then make sure, on the new computer, when you plug in the same drive, it have to give you to same drive letter D.
I am pretty sure you can use the File -> Open Catalog command and select the new catalog from the connected drive. You don't need to rename it.