if you do this from an external hard drive, it is slower. so i would recommend a lighter system, linux mint xfce, or lxle for instance, depending on how light you need to go. the appearance of these can be altered.
the installation process should be simple, there are main options, like replace Windows, use free space, something else etc, but instead of choosing an automated method, you may need to use the something else option for manual partitioning. in the partitioning tool there will be a place to select the hard drive, so check it is the same size of external hard drive, will probably be referred to as "sdb". make these partitions:
[swap]
-- filesystem: swap/linux swap
mount point: no mount point
size: same as ram + 1/2, unless you have 4gb or more, 4gb should be fine.
[main system]
-- filesystem: ext4
mount point: /
at the bottom you will get a choice of where to install the boot loader Grub, this is different for you because the internal hard drive is possibly completely unusable, therefore you cannot install to that drive. you need to either install to a normal usb flash drive and always have it connected to boot the external hard drive, or install to mbr of external and choose it as default boot device in bios so it loads the system.