Which vine or other plant would work best?

I have an old chain link fence still in good repair that I'd like to preserve, but gain some privacy from my neighbors. The fence is located on a mostly sunny side of my house, and I live in Miami so heat and rain are major considerations. Any suggestions on what kind of vine or other plant would thrive best for this type of application? Also if I plant a vine what can I do to ensure that it will not spread and destroy my neighbors St. Augustine grass?