I want to re-install my MiKTeX distribution on my computer. What would be the most advisable way of doing it? What is the process of uninstalling the previous version safely and installing the new one?
MiKTeX has no own button or menu to uninstall it.
To be sure to uninstall MiKTeX completely just use the "Windows Programs and Features (Add or Remove Programs)" control panel to delete MiKTeX.
Then reboot your computer.
Then download and install the new MiKTeX from the homepage http://www.miktex.org.