The DLC will start where you fought Mohg, probably by interacting with Miquella's egg. If you can access that, you can access the DLC. You don't have to start over, do your thing.

As far as we know there will be some separate scaling from the rest of the game so if you're leveled for the endgame you should be able to still enjoy a degree of difficulty. However, it's hard to know if this is exactly what Miyazaki meant in that interview.

Personally I'm going to do at least 1 dlc run by starting over, and 1 by picking up an old character.

Keep in mind that you might have to jump through some hoops before you start the DLC regardless of starting over or not.

Every other FromSoft DLC takes place right before the end of the game, essentially. They all have end-game level enemies and weapons/spells. This one, however, will apparently be different, although having only played Sekiro for an hour or so I am uncertain what they mean. So it sounds like you could go either way. Personally, I'm going to start it with my current character and see how that goes.

Create a new character. I don't know if your level will be important, since I heard, that the DLC uses a Sekiro like leveling system.