1. Direct one's hopes or ambitions toward achieving something: "we never thought that we might aspire to those heights".
2. Rise high; tower.
If you’ve never written a screenplay and you’re in the process of going to classes, buying books on the craft, reading scripts, and plan on writing one. Then you’re aspiring to do it.
You haven’t done it yet.
If you already write screenplays and write constantly and consistently, you are no longer aspiring to be a screenwriter. You are one.
You’re probably aspiring to be a paid professional with produced works and credits to your name. This is no doubt the goal for most screenwriters and of course the joy of the actual writing process itself.