The first thing you need to do is download the following two files:
(Even if you only plan on editing MXML files, you need both of these. The MXML syntax depends on the ActionScript syntax, because MXML files can include ActionScript code.)
(Thanks to Abdul Qabiz for these files.)
You need to put these files into vim's syntax folder. On Ubuntu 7.10, it's /usr/share/vim71/syntax. (Make sure you install vim-full, using sudo aptitude install vim-full ... for some reason my installation of Ubuntu came with vim-tiny, which doesn't support syntax highlighting.)
Once that's done, you need to make sure the following lines are in your ~/.vimrc file:
au BufNewFile,BufRead *.mxml set filetype=mxml
au BufNewFile,BufRead *.as set filetype=actionscript
Now ... the next time you open up an mxml file in vim, it'll have syntax highlighting for you. Excellent. My development environment just got a little bit better.