You need to rank up to level 30. Each level up requires 100x your current level (except the first level), so in total you will need roughly 45,000 XP.
You get experience by playing games. A multiplayer match earns 25xp (always). For normal games, you get xp for the following items:
Stars: 0 = 5 XP, 1 = 9 XP, 2 = 12 XP, 3 = 20 XP
Boost monster: Common = 3 XP, Rare = 7 XP, Legendary = 15 XP, Mythical = 20 XP
Evolved monsters: Common = 1 XP, Rare = 3 XP, Legendary = 6 XP, Mythical = 10 XP
5 XP bonus for evolving a monster for the first time.
Thanks to Mike Marcelais on TA for providing a breakdown of the XP system.
After you have gotten all of the other achievements, boosting on multiplayer is the easiest. You are guaranteed 25 XP per game no matter what and do not have to worry about any nasty 30-second ads. Essentially you can just keep starting multiplayer games and making a single match and let the clock run out. This will earn you roughly 25 XP ever 90 seconds, or 1000 XP per hour. This is roughly the same rate that you will earn while repeatedly playing level 15 of Forest to evolve all the Mythical monsters for the "A World Much Sweeter" achievement. At these rates, it will take you a minimum of 45 hours to complete this achievement and the game.