Leveling Lightning
Lightning, available in the Lightning: Requiem Of The Goddess DLC, advances differently than any other character or tamed monster. Her play experience is entirely self-contained, and consists of a pair of battles that she can repeat endlessly. The core mechanic at play is that she earns CP even if she loses. As a result, if you have trouble at first, you will eventually become stronger until you are able to claim victory. (It is virtually impossible to lose at level 10, should it come to that.)
Lightning gains CP after each battle, but unlike other characters, she never spends this CP. Hitting certain CP thresholds will automatically level her up. It is entirely possible to gain more than one level at a time after a particularly well-fought battle. You will also gain 10,000 CP for winning the second fight, which will immediately unlock every level. Note that Lightning's battle will end unless you earn five stars on the first fight. You will generally earn a maximum of 100 CP from this fight, so the key to rapid advancement is, as you might expect, rapid victory.
It is of particular note that Lightning does not have to fight at her current level—she can adjust her level down to make the battles more of a challenge. This has the added effect of increasing her score multiplier. Winning at low levels is the key to really high scores. Note that level 0 is not unlocked through CP accumulation, but rather is a reward for winning the second battle for the first time. This is a challenge mode with (relatively) low stats and less abilities even than those known at level 1.
Lightning's Roles
It is worth noting that Lightning's roles differ from those found elsewhere in the game in more than just name. She has two Ravager-based roles, and none based on the Medic. (Healing is available in the form of 5 Potions and one Elixir per attempt, and use of her Legion of One Feral Link ability will also restore a small amount of HP.) These roles don't play exactly like the roles they are based on—the Shaman, for instance, can keep a chain gauge from dropping much better than a typical Ravager, while the Mage is more traditional in this respect. It's especially worth noting that the Conjurer and Sorcerer both fill their ATB gauges incredibly slowly—consider switching to them when you have an ATB refresh available, or doing so when Lightning's ATB gauge is already full from another role.