Will writing fanfiction improve your writing ability?

Writing improves your writing ability. It is a skill that must be practiced often and diligently just like anything else. It doesn't matter what you write, as long as you write. It is also good to study the writings of others and read/learn about writing and grammar. These will help you with the craft itself. Studying good writing teaches you how to write well and studying bad writing teaches you what not to do. Grammar and writing basics will help you with the language itself. No matter how good a plot is, readers will not suffer through a story they can't read and understand because of bad grammar.

As you learn the craft, keep writing and applying what you've learned. It's okay if you don't write well as long as you continually seek ways to improve where you fall short and then do it. Once, you've got the basics down, you'll find yourself developing your own style. Write whatever you want: prose, poetry, scripts, novels, essays, comics, fan-fiction... as longs as your writing you'll improve. The key is to write as much as possible, especially when it's bad and even when it's good. ;)