I am currently dungeon mastering a level 5 campaign, and the first session went... Interestingly.

It started with me saying "what do you want to do?" Because I didn't have anything planned. There we have a dragonborn bard, a fairy druid, and a goliath barbarian. My bard and druid wanted to go to the governor's keep, while my barbarian stayed back and "kept watch" in the town square. When they got to the keep, they rolled a nat 20 on persuasion to get the guards to not confiscate their weapons. They then enter the governor's office. They cast charm person on the governor, and he dismissed his guards. They locked the door, and stabbed the governor to death. Meanwhile, the barbarian is coming up to the keep. He goes up to see the governor, hears a commotion, and breaks down the door. As he walks in, the bard activates his draconic flight, and they both cast invisibility and fly out a window to the nearby forest. Now, the door is broken down, the governor is dead, and our barbarian is standing in the room. 

The barbarian has sworn revenge on them, and the bard and druid are plotting something. By question is, where do go from here, and how do I keep both groups on the same level?

You stop this campaign and have a session 0 where you all talk about expectations and playstyle and what everyone, as a group, wants to do. Then you start a new campaign where everyone is on the same team. Because, honestly, this one is about to fall apart anyway, so you might as well make the next one work better.

Side note: Charm person is not mind control. Just because the governor is friendly, doesn’t mean he will dismiss his guards. 
Also, persuasion checks are not mind control. You are well within your rights as DM to say no amount of persuading will make the guards be willing to allow armed people inside. In fact, you don’t even need to let them roll. You can just say “they won’t allow you to go in there armed.” The players do not decide when to roll, you tell them if a roll is needed.  

Quote from Xalthu >>

You stop this campaign and have a session 0 where you all talk about expectations and playstyle and what everyone, as a group, wants to do. Then you start a new campaign where everyone is on the same team. Because, honestly, this one is about to fall apart anyway, so you might as well make the next one work better.

Side note: Charm person is not mind control. Just because the governor is friendly, doesn’t mean he will dismiss his guards. 
Also, persuasion checks are not mind control. You are well within your rights as DM to say no amount of persuading will make the guards be willing to allow armed people inside. In fact, you don’t even need to let them roll. You can just say “they won’t allow you to go in there armed.” The players do not decide when to roll, you tell them if a roll is needed.  

 I understand these points, but they rolled a nat 20, so I felt like I kind of needed to let them do this. Also with charm person, they said that they needed to talk in private with the governor, so I felt like that would work. Also, I want this to go this way, sort of, but I want them to be on the same team. I think this dynamic is funny, but can't understand why our chaotic neutral barbarian is so against joining them.

Yeah you really need to stop and have a session zero. The barbarian is against joining them because regardless of what alignment they chose the *player* doesn’t want that sort of campaign. If the other two players, and you by the sounds of things, do want that sort of campaign you need to all agree to that and give the barbarian player a chance to leave because they’ll end up miserable if this goes on