Do you have to kill people in a private online game?

You don't "have to" kill people, but the game is designed to make it very easy to kill people. Say you're driving down the road, your motorcycle hits a corner too hard and careens into the sidewalk running over pedestrians. Boom you've accidentally killed people, their blood on your hands.

The entire game will make killing people nearly unavoidable as time goes on, even if you only play the multiplayer, but by no means are you forced to do so.

Killing people is completely avoidable in GTA Online if you want to play the game that way but you will obviously be limited in the content you can take part in, since the game is about heists and murder. There is even a "passive mode" in GTA Online which specifically prevents you from pulling out a weapon or being harmed by the actions of others (it won't stop you driving over people, though).

If you want to play GTA just to drive around and admire Los Santos, there is nothing stopping you from doing that, even in single player mode (once you're past the initial few story missions).