How do I change the background image of the main menu?

Here's a pretty detailed guide on exactly how to do this, including screenshots and video tutorials.

http://elusive-teliko.blogspot.com/2011/04/customise-background-images.html

I haven't tried it, but one way to do it should be the following:

• Get a program that can read and write .mpq files, which Blizzard uses to store game resources. For example this MPQ editor.
• Make a backup of your Starcraft II/Battle.net/Battle.net.MPQ file.
• Open the Starcraft II/Battle.net/Battle.net.MPQ file in the editor.
• In Battle.net/Shared in this file you see lots of .dds files with cryptic names. These are the images displayed in the game client.
• Extract these .dds files.
• Get a program that can display and encode .dds files, for example ATI's The Compressonator
• Find the image you want to replace. If you are searching for a background image I would recommend to to start with the largest files. Maybe there are also thumbnail programs out there that support .dds which could be useful to find what you are searching for.
• Encode your own picture as .dds and give it the same name as the one you want to replace.
• Replace the original picture with your newly encoded one in the .mpq file.

With this modified .mpq file, Starcraft should display your modified picture.

Note that you are probably not allowed to modify game files. It could be considered hacking and might get your account banned, or it might interfere with future upgrades. I don't recommend that you do this.