"My goal is to become an aspiring writer."
An aspiring writer is someone who hopes to be a writer one day. So the goal is to become someone who hopes to be a writer.
I am all for trying new things out to see what I like to do.
I would suggest though, that if your goal is to try to try... you probably won't.
"My goal is to become a successful writer."
(Please leave a comment with your own examples.)