apt dist-upgrade is different from apt upgrade in that it allows packages to be removed, and expects the user to carefully review the list of packages being removed to make sure none are important.
If you are on debian stable, there is no reason installing vlc would require packages not in debian stable. Odds are you have added third party repos or repos from other versions of debian, i.e.
frankendebian. Once you do stuff like that, you're kinda on your own to resolve it as the OS is in some unknown state.
If you can't get a TTY with ctrl+alt+F# then you could try logging in via ssh. Or booting single user mode.
To attempt to understand what happened, you could start by sharing "apt policy vlc".
You can also take a look at the apt log to see what you removed. /var/log/apt/history.log
You can take a look in /etc/apt/sources.list.d/ to see if you added any 3rd party repo there.