If I was...

Depending the kind of art that you create, the look of your site will look different.
Maybe the set up and organization of the page would be very similar to other kind of artists but the imagery will speak itself. I did create some examples on my personal facebook to explain this fact easier:

If I was a illustrator:
(who I hope I will be one day) I would use as a profile picture one of my illustrations, finished and looking nice, but also I would use some other illustration on top, in this case I used one in which you can see  a finished image but still with papers behind, like if I didn't finish it yet, just to show clearly that are my drawings, and kinda "how I make them" sensation, even if is something you can't see on the pic.

If I was a tattoo artist:
Maybe the imagery I have used here isn't the best but just to make an idea... the profile picture would be an own tattoo design (preferably with color) and on top I used a picture of an old couple completely covered on tattoos, maybe I could use a large own design as well, the point is make sure this imagery is directly related with tattoo, so once you see this profile you don't have a doubt about you're checking a tattoo artist!!

If I was a photographer:
Clarifying the fact of being a photographer I upload a picture of myself with my camera, (wich is very clear about the sent message of: I do take photos), but also I used a large nice picture of some crazy buildings of my city, giving an example of "my work" (actually this picture was taken by my sister).
But with this two images together isn't easy to connect "the girl who take photos" and the nice landscape as taken by her, like an example of her job.

If I was an street artist:
I would never ever put my face on my profile!!!!
But instead use examples of my work: as a profile picture I used a stencil painting, something cool and easy to fit in a small space as the profile pic, probably something that the artist himself use to draw, like cheo's bee (this is just a nice graff I've found around Bedminster).
And then on top I would use a big piece ( I just used the pic of my 1st trial with sprays), maybe a big lettering piece with the name on it...

The point (again) I want to bring up is: It doesn't matter the kind of art you create or the platform you are using to promote it, the important point of it is the imagery you  are using in, how you are showing to the rest of the people what are you doing, so this imagery meant to be directly related with you and your work (pictures of your works, yourself working...), so no one will have the doubt of what they found when they clic on your page.