Building a consistent and professional brand is a key piece to establishing yourself. If you have a website (and you really should), you probably already know this. But your brand should expand beyond your website, especially if you want people to actually find it. A key place to bring your brand is social media, like Twitter and Instagram.
The first step is to bring a consistent username across your social media accounts, ideally your website minus the top-level domain (.com, .net, .me, etc). For example, my website is 'designsbyhall.me', and my username across social media is 'designsbyhall'. You pay good money for your domain name, might as well get your money's worth out of it. If you already have social media accounts (and I'd be surprised if you didn't), most have very simple ways to change your username. Also, be sure to use the same picture across your accounts, like maybe a photo of you from your about page.
Another key point of your brand consistency is your email address. You definitely don't want an email address like 'email@example.com' to be a reflection on your brand. You probably don't want one of those free email accounts either. I've found that it's worth paying for your own custom email address, it shows you're professional and trustworthy. I use Hover for my domain name and email address. They're very reasonably priced, and I highly recommend them, but there are other options out there (just stay away from GoDaddy).
Curate your Content
So now that you have your social media accounts under your brand, and your very own custom email address how do you present yourself? Be professional, but also don't forget to inject a little of yourself into what you present. You don't want to come across as a robot (unless you actually are a robot), but you do want to keep yourself in check. Stay away from extremely polarizing topics like politics, religion, and which mobile OS is the best. Also, don't swear.
You also want to focus on the kinds of things you share via social media. You definitely want to focus on your design work, but also show the world that you are human. You don't want to share everything, but you maybe want to pick one or two things that define you (again, excluding politics and religion).
It's not going to be something that will become comfortable overnight. When it doubt, err on the side of caution. Over time you will eventually fall into a place where you are comfortable. Like anything it takes practice.