Sadly JavaFX doesn't link that easily to an sbt build. You need to set your JAVA_HOME environment variable and do modifications to your build file.
Here I have a repository where this is set up. The important bit if you are using an sbt build rather than a scala build is this one:
unmanagedJars in Compile += Attributed.blank(
file(System.getenv("JAVA_HOME") + "/jre/lib/jfxrt.jar")),
fork in run := true
The reason for this is that jfxrt.jar is the archive containing the JavaFX runtime and it is not included in the classpath of an sbt project by default.