CV's are somewhat controversial among freelance translators. I stand by my old opinion that it's not demeaning to have one and call it one. Lawyers, politicians, professors, CEO's and all sorts of people have CV's, so a freelance translator also can. If you're interested in the subject, Marta Stelmaszak's Lesson 57 has a compendium in the form of a massive free e-book with very helpful tips from people who have actually sifted through many of them.
On the other hand, in a job line in many ways similar to ours, lawyers are generally using bios, which is short for biograms (tiny biographies). At least 60% law firm website traffic goes to bios, possibly because few other things are either interesting or understandable to clients. Besides, getting 'conversions', selling etc. is largely about selling a story. And a bio is by definition a story.
If you go to the popular lawyerly blog at Lawyerist.com — which contains a massive resource of very accessible and very transferable marketing advice (again, because of the similarities between our job lines) — and if you enter 'bio' in the search query, you will get about 90 results. Curiously, visual CV's mentioned in Randall Ryder's advice/opinion post there are not unheard of among translators, but I don't see many bios out there. And they are really good replacements for a classical CV on a translator's website — and possibly only there for lawyers, though not really for us translators. We could probably use them more extensively than lawyers, notably in all of those directories we're listed in where a free-form profile is allowed (e.g. Proz.com's About me section), perhaps in brochures.
Using a bio allows you to be narrative and engaging, even though you lose the the benefit of a more structured presentation with key facts arranged chronologically or functionally or enumerated in a bullet list. However, you also get a chance to leave the CV contest behind. CVs fit rather closely in the paradigm of competitive variables (price, deadline, and credentials), bios break out of that scheme. Also, where there is no CV — and some of us have already pulled theirs due to the threat of scammers stealing them and replacing contact data with their own — there is less vulnerability to identity theft risks.
What else? If you'd like to give it a go or at least 'a think', I'd suggest you take a look at field experts from your areas of expertises. Lawyers, doctors, actors, writers, other people, how are their bios written? What do they have on their biographical or at least profile pages? What can you adapt from there? Can you use the knowledge to enable them to relate more easily to you? Bios are certainly not the only thing you can adapt once you get there and take a good look.
You will probably benefit the most from having a freeform bio, as opposed to the restrictions of a classic CV, if you're a native speaker writing in your own language (non-natives will find it much harded to keep up with you in creative pursuits than in CV writing) or somewhat of a copywriter/transcreationist, or both.