To level your character, 3 of the same character (and level) are required. Use the upgrade screen to combine these 3 characters into 1 character from the next level. From level 7 onwards, only 2 cards of the same type & level are required to level up. For example, 3x level 1 Skeleton Deathknight creates 1x level 2 Skeleton Deathknight.
As you upgrade a character their Attack Power & Health Points increase, sometimes even gaining a new ability as well. To open the Upgrade Menu click on the "Upgrade" button on the Deck Menu. The upgrade view is scrollable up and down. Any characters you have in Royal Rampage will show on the upgrade screen if you have 3 or more of any particular level. If you only have 1 of a character it will not appear in the upgrade view. The numbers above the input boxes are the character levels. The numbers underneath the input boxes is the amount you own in your account. If you own characters the balance text is green. Note you cannot input a number for level 10's because level 10 is the top level, however you can see your balance for level 10's. For speed and ease you can upgrade multiple characters in 1 upgrade. Enter how many of each character you would like to level up in the input boxes then click the "Confirm" button. The view will change to show a review of the characters you are choosing to upgrade. Click "Confirm" again to make the upgrades happen.
To save you time finding the cards you want to add to upgrade you can use the Filter options. Click the Filter button at the top of the screen to open the Filter options. You can select just the rarity or just the level or just the character race. You can also select a combination refining your search by rarity, level and race.