Why should you play Lillia?
In a Nutshell: Fast Clear, high mobility, big burst of damage, hard to catch, CC potential.
Lillia has an excellent clear in the Jungle, which gives you the potential to be everywhere on the map, without losing on farm. Her MS makes her very hard to catch in team fights, so you can make her a very strong AP damage dealer, or a bit tanky F1. Her CC on r can change the fate of many fights, so she's a strong flex pick to different teamcomps, or just on SoloQ as a carry.
Runes
Summoner spells
Bans
As Lillia, you want to avoid something that can easily catch you or CC you. A pretty good ban is Nocturne, which can easily dash in you, negate your spell, and CC you for pretty long time
Abilities Max order
Items
Playstyle in a Nutshell
Early game:
In Early you Should avoid pointless fights or ganks, you want to be in tempo everywhere. Focus on Clearing camps and snowballing. Gank only when you see that you and your team will get a clear advantage in fights. Avoid solo fights, always look for teammates, to make it 2v2, or 3v3. if the enemy takes an objective, and you don't have teammates to help you, just give it, and go back to clearing camps.
Mid Game:
In Mid game you can start looking for more skirmishes, you will be much faster and harder to catch. You can play more aggressively, start doing solo objectives, or start invading enemy Jungle with your team. BUT if you will be behind in gold, just play like in the early phase of the game, and focus on the farm and clear free kills.
Late game:
Lillia gives much value to your team, so you need to stick to your teammates a bit. Look for random E in the enemy team, and if they will engage, you can easily disengage them with your ultimate. If you have Flash, look for random engage with it and sleep everyone you will hit. If you don't have flash, just look for a safe opportunity to engage anyway, but avoid random CC that you can get from an enemy.
Last words
Remember to be careful when the enemy team has champions like Nautilus, Thresh, Leona. They can easily catch you, and kill you with a team. If the enemy team has many champions who can catch you, consider building Zhonya.