To achieve the maximum fame, it must be level 33 (Demigod). You can still gain more levels in fame even though the achievement calls it maximum. This will require you earn 756,000 fame experience points. You earn fame by defeating Champions (rare monster who have specific names). They're usually either bigger than other enemies, or glowing (or both). You should see a couple of them a floor, especially in higher numbered floors. As you level enemies level, as will the fame you get per Champion monster killed. You'll be getting around 10k for each Champion you kill once you're a specific level.
Each class has a specific skill that grants you more experience and fame. You will want to max out right away to save time. This skill is called Adventurer. You'll have to kill roughly 200+ champions to get the max fame level. You can check how many you have killed in your stats screen. You will be able to get this skill, with any class, at level 10.