If you want free Wowhead Premium, then you've got to contribute to the website, because all members with more than 25,000 reputation will get it for free.
How to get Wowhead reputationFirstly, you obviously need an account. Sign up with you e-mail address and log in. Once you're in, just start contributing. Some people prefer to make good comments on the pages of items, NPCs, factions etc., while others prefer other methods to get reputation.
The easiest way to get to 25,000 rep is probably by taking a screenshot of 2,500 different items, mounts, or NPCs and upload them to the website. Each screenshot will get you 10 reputation each, and while 2,500 screenshots sounds like a lot, it will go by pretty fast.
Once you get enough rep, the Wowhead Premium will be permanently attached to your Wowhead account, so after this point you won't need to contribute to the website anymore unless you want to.