The Genre field is a simple string. While you can add semicolon separated values, the standard views don't recognize them as anything special - its just a long string.
You have two choices essentially:
1) modify the views to use expressions to split and present multiple values
2) create your own Genres (note plural for distinction) library field, which is of type List.
Using the native list types has many advantages. The downside is that your own fields are not standard, and external software won't utilize these tags as you would desire.
Here's a big, excellent thread about these matters:
http://yabb.jriver.com/interact/index.php?topic=57477.0
