How do I modify the ci command in vim

The vim LaTeX box plugin adds this feature. If you didn't want to use the rest of the plugin, you could just look at the source and see how it is done.

I think some of the other LaTeX plugins for vim probably have that feature too. There are at least four major ones, the LaTeX box one is the one I'm most familiar with.

I don't think so; you'll have to have some way of telling vim whether you want to change between the dollar sign forward or back, and I don't think that it can be programatically determined.

You can, however, do a "cf$" to change everything from where you are on the line until the next dollar sign, or "cF$" if you want to go to the previous dollar sign.