You can install Ubuntu into a USB drive like it would be installed into an internal drive (but to a USB drive).
After the installation it is possible to install other kernels, also a custom kernel, if you find a ppa or a deb package for it.
If the computer is running in UEFI mode, and you want to boot directly from the USB drive, it is easier if you remove the internal drive before you start installing. Then the USB drive will be regarded as an internal drive, and the installation will be easy. I would recommend at least 16 GB for such an installation (a fast USB 3 pendrive would be fine).
Live or persistent live system
A live drive and a persistent live drive will make the computer boot from the kernel, that comes with the iso file. You cannot make it boot from another kernel, and it is very complicated to create an iso file, so I recommend that you use an installed system.
See these links and links from them for more details,
Short description showing differences between installing an installed system and a persistent live system
Try Ubuntu (Kubuntu, Lubuntu, Xubuntu, ...) before installing it