Avoid spamming abilites before your first back. You don't have much mana, so if you spam Q and E you WILL run out of mana. Let the enemy push the wave into you so you can freeze outside of your tower. Poke them with E+Q as they overextend to farm their minoins. Back around level 4, TP to lane if necessary to catch a minion wave. At this point you are starting to become powerful.
Try to get them low and then use your R as an execute.

With your mythic item built, you should be able to duel nearly anyone. If your lane opponent is a tank with sustain, try one of two strategies:
1. Constantly shove the wave and roam.
2. Freeze the wave and deny the tank experience and creeps.
If you haven't fallen behind, you will win most skirmishes you are part of.

Show up to all objective fights. Lillia isn't a good split pusher. You need to have passive stacks before the fight starts, so look to hit jungle camps, minions or neutral objectives before. The QFlash + R + Zhonya combo is very powerful and can win you games.