You don't need to do anything, it'll work without Windows, and grub should auto update with your distro when you do a major update (such as Kernel update)
You might want to manually update grub to remove the Windows entry just to keep it tidy. On mint it's as simple as:
sudo update-grub
It'll scan your installed kernels and other OS. If windows is gone, it will no longer be detected and disappear from the boot list after running this.
If you've set up a default OS at boot (like Windows) you might want to update the grub config files. Thats as simple as editing /etc/default/grub and setting:
GRUB_DEFAULT=0
Where 0 is the first boot entry.
You can also use:
GRUB_DEFAULT="saved"
which will remember the last selected item and boot that. Bit redundant if you're going down to 1 OS.