In ‘3 qualities of successful Ph.D. students’, Matt Might writes on cogency:
Generally, grad students don't arrive with the ability to communicate well. This is a skill that they forge in grad school. The sooner acquired, the better.
Unfortunately, the only way to get better at writing is to do a lot of it. 10,000 hours is the magical number folks throw around to become an expert at something. You'll never even get close to 10,000 hours of writing by writing papers.
Assuming negligible practice writing for public consumption before graduate school, if you take six years to get through grad school, you can hit 10,000 hours by writing about 5 hours a day.
Of course, you're not doing a Ph.D. to become an expert writer, you're doing it to become an expert in your field of research. But the point can be a shocking one: when I was early in my Ph.D. my realisation was slow that my writing was, at best, very average. Then consider that an Australian Ph.D. will take on average only 3.5–4 years after a four-year bachelor; we skip the important learning experience of doing first a Masters project.
Writing a blog isn't the only way to practise writing. Document your code as if others were to be using it; write notes on papers you read as you read them, don't just file them in a bibliographic database for ‘one day’ writing a literature review. Joining mailing lists and asking for and offering help and advice online will quickly make you realise how difficult it can be at times to make yourself understood through the medium of text online.
I also find that reading a lot helps my writing, but I'm pretty ashamed of the paltry number of books that pass by my bedside table these days.
For future reference (for myself, mostly), here are some tools I've recently read about for improving your writing without expending any time to do so:
These certainly won't do much to improve your writing, but if they save even one silly doubled-word mistake in your thesis, they're worth it.