Try to look up any book on the is successful singer-songwriter and a single living off at, usually, the title of the book will have something in it about the music business. In the music circles, they are definitely clear that while music is a noble art form, but one does have to master the business side of it if one is to really succeed making a living with it. When it comes to being an artist and trying to sell your paintings though, the "art business" mentality is a relatively recent one. You really should art business books though. They're all kinds of novel ways available today of promoting and selling art online.
The first rule of promoting and selling art online is to really promoted online. The best way of doing that of course, would be to start your own website and blog. You really need to design the website properly though.
It has to have a name that people would think of when searching for work of the kind you do, and it has to have great search engine optimization. These days, in addition to what search optimization used to involve in the past, you have to put in a few pictures, videos and other such stuff to keep it colorful.
But of course, since you're selling art online here, there are doubtless going to be plenty of pictures.
The other part to optimizing your website to be popular with the search engines, would be to make certain that there is plenty of content on there. Read up about how to ingratiate your website to Google. Google needs to see plenty of text that is relevant to your subject. Try to start a blog about your view of art, the latest art techniques, try to start a little series of tips on drawing and so on, and you'll be pretty popular in no time.
The thing is, the more people get to know about you and your website, the more they'll come to hear about your art. You need to give the whole world plenty of reason to stop by your website for artistic purposes. There are things you can do other than write how-to articles too.
Have you ever noticed how good instructional videos on YouTube get fanatic amounts of following? You could create videos yourself about how to paint. You can be sure that a great following like this on not only YouTube but also on Twitter and Facebook as well, will be invaluable when it comes to selling art online.
Not to mention, you could really be a presiding authority on a couple of popular online art forums. That would establish you as nothing else could.