After defeating the Elite Four and becoming the newest Kanto Region Champion, you can rematch the Pokemon League whenever you would like. When facing the Elite Four again, each member will use the same Pokemon as before (but at higher levels), as well as adding one Alolan Form Pokemon to their team that matches the type of Pokemon they specialize in using. You will continue to earn loads of Experience and Money when you rematch and defeat the Elite Four.
When it comes to Bruno, the second member of the Elite Four, he will still specialize in using Fighting Type Pokemon and Rock Type Pokemon, continuing to battle with Onix, Hitmonlee, Hitmonchan, Poliwrath and Machamp. However, he will also use a Golem in its Alolan Form in this battle. This time around, all of his Pokemon will be at Level 62, with the exception of his Machamp that will be at Level 63.
Just like your first battle against him, Flying Type Pokemon actually work best against Bruno's Pokemon Team. Flying Types do serious damage to his Fighting Type Pokemon, and they don't take much damage in return from their Fighting Type Attacks. When it comes to Bruno's Onix and Alolan Golem, however, Ground Types and Water Types work best against them. Water Types can really hurt them, but Ground Types are recommended since Alolan Golem is part Electric Type and can hit Water Types hard in return. Finally, make sure the Pokemon you do use against Bruno are at Level 65 or greater, as this will allow you to match up well against him and the rest of the Elite Four.