If you want the background on your userpage, adding it to a CSS page might not be an option. I imagine bloating a wiki's CSS with specific styles for each userpage (in case more users request that) would be a burden for admins.
A more self-service solution would be to use JavaScript to detect when we're viewing a userpage, and include a CSS from a subpage from the current user page, that way users can use that stylesheet (that only the own user would be able to edit, apart from admins) to style the entire user page as they want. That, however, has more security implications than external images, since any user can exploit various CSS vulnerabilities from old browsers and load arbitrary CSS to people viewing their userpage.
The ideal solution here would be a MediaWiki extension that provides an interface to allow users to upload an image that would be used as a background image for their userpage, an image that's probably not stored as a normal file and that only that user is able to change. Admins should probably have the ability to delete any background image that they consider inappropriate. There doesn't seem to be a similar extension, though.
If external images are allowed on that wiki, or you can use an uploaded image from the wiki, you could also place it on the page, and then put the rest of the page content on an absolute positioned div on top of that image. That would give it a sort-of background image functionality, with their problems: you can't make the background repeat, it would make the content fixed-width, etc.