You need an image with sufficiently large dimensions. Width generally won't change, though it'll always look different depending on the screen resolution of each viewer. An image 1920 pixels in width should look fine (in terms of not leaving empty space) on most screens, though if you want to be safe, you may want to consider one that's 2560 pixels wide.
Height, however, will contract or expand depending on the contents of the page. The area covered by a user-defined background image on a minimalistic profile page (without long visitor messages or biography) is typically around 700 pixels tall. This will drastically increase on profile pages with a lot of content, such as long visitor messages or a high amount of groups joined. For your profile page in particular, you'll likely want an image at least 2700 pixels tall.
Once you have your large image, you may want to turn off background repeating. You can do so by clicking on the colour palette next to the "Page Background" option, which will pop up a window as seen below:
Lastly, please remember that only page background under the "General" section will take effect for the whole page. Backgrounds defined under other sections will only correspond to certain modules such as the left panel or the visitor messaging panel.
You're welcome. I'll consider the question answered. Let me know if you've still got more and would like the thread reopened.
An addendum, if I may: using one exceptionally large background image covering your entire profile page isn't generally the best option. Quality, large dimensions, and fast loading times―you can only pick two out of the three. It's best to use a small image that connects well like a pattern and let the built-in CSS handler repeat it.