Has anyone run a villain campaign? Thinking of doing this at some point but not sure of a story. 

I only made it as far as the planning phase. Every character had their own motivations that didn't tie into eachother at all. And they are evil, so they didn't have any reason not to abandon or kill other players if they had nothing to gain. Not to mention one character was the definition of a murder hobo.

Evil campaigns have certain challenges. Make sure to have players come up with their characters together under your supervision in a session zero. Then you can plan a campaign based off of their ideas.

Yup. Actually with October and Halloween coming up I've asked my players to roll new characters for an evil two session run.

First. You need rules. Players are not allowed to kill each other. They are a team who knows each other and trusts each other.

Second. Need a decent story hook

Third. They are on the story hook so the world around them has to fit in the story. Write a world where they can cause some chaos in. A place for them to have fun.

This is where imagination comes in. With my group I'm not writing a lot. Just bullet points and letting the players have fun in the world. But that's only cause it's only two sessions long.

Though there is no such thing as an evil campaign.

In the group I play with regularly, we just wrapped up a campaign in Tal'Dorei. Each of us had two characters, a main party good guy, and and evil party bad guy from the hobgoblin empire in the south.

Things started out with the good guys, and then every so often we would cut to the evil party doing stuff. And so for some events we saw it from both sides, like when the evil party razed a halfling village and the good guys to set off a magic bomb to feel a void dragon, and then the good guys are sent there with "oh God, what have they done"

Of course said void dragon betrays the evil group so they end up having to team up with the good guys.