Can i play on one on one with a friend in Fortnite?

Yes you can in Playground! Once you are in the game with your friend, you can hit escape (on PC) or whatever button it is to open the options menu. On bottom you should see a button that says Team Select. Have one of you set your team to Team 2 and voila! Now you are separate teams so you can kill each other until your blue in the face.

Just quick notes about playground mode. First, the material gain is increased dramatically. It should take no time to get max of all types. Second, the ability to open the glider anywhere is in effect. So don't worry if you fall from a great height. Other than that, it will be the same as regular game play.

As @Niffler said, the same can also be applied to the Creative mode, however this mode is quite different from playground.