Just like on PC, you have to be a level 25 to start your 10 preliminary matches to get put into your skill rank. 

Along with competitive mode, the new Overwatch update brings some new minor fixes to the gameplay: 

The first fix is that the player will have “increased accuracy during off-center and diagonal stick movement. " For all you lovers or haters of aim assist, you now have the option to adjust it to your liking. Avoid this player option has been removed. McCree is now available to use in Practice, Against AI and custom games. McCree has been nerfed a little bit. Instead of his ammo doing 75 damage, it will now be decreased to 45. McCree’s reloading time decreased significantly. It will now take 0. 3 seconds instead of the 0. 75 it took originally. Widowmaker’s Widow’s Kiss scoped shot has been decreased from 15 to 12.  

The update is available now for Xbox One. There has not been a release date for the PlayStation 4 update.