Kate is being tried in California, despite having committed her crime in Kansas. In the United States, murder is a crime at the state rather than federal level and murder trials are conducted in the state where the crime occurred. From what has been shown in Lost, Kate has committed no crimes in California and so would not be charged there. Even if her flight from prosecution was a federal crime it would most likely be prosecuted in the federal court serving Kansas and not in California. It also likely that the state charge of murder, being the more serious offense, would take priority over any federal charges she might face.
The courtroom scene contains numerous errors in how proceedings are conducted, ones that are noticeable to those with even a nodding acquaintance with American legal proceedings. Among them are the fact that the prosecution presents its case first and then the defense.