As a copywriter, you're expected to perform miracles - write something creative while adhering to logic, rationale and process. In most cases, the latter takes precedence over the creative bit, therefore taking away the copy's USP and personality.
Basically, get in the zone, be brave and practice. And if you can't get in the zone, get out of every zone. Clear your mind and wait for that flash of inspiration.
Don't be afraid to fail. Push the boundaries and see what happens. It may not make through the approvals process, but you tried and that's what drafts are for. So experiment, it's better than producing lacklustre, trite copy.
Write. Keep writing. Write anything from tweets to novels. Play with language, ideas and platforms. Write until it becomes as natural as checking Facebook the minute you hit snooze.
My biggest bit of advice that all writers and potential writers find useful is when you're not writing, read. If you need to write something akin to a news piece, read the Guardian online. If you need to write a play, read unabridged Chekhov. If you need to write product pages, visit the site of your favourite retailer. Immerse yourself in the copy and get your brain thinking like that, then the words will just flow.