You can try replacing your `def run(self):` function in your script for the one below which should do something you want:

    def run(self):
        """Run method that performs all the real work"""
        # show the dialog
        self.dlg.show()

        # Get all loaded layers in the interface
        layers = self.iface.legendInterface().layers()
        # Create an empty list which we can populate
        layer_list = []
        # For every item (which we call "layer") in all loaded layers
        for layer in layers:
            # Add it to the list
            layer_list.append(layer.name())
        # Clear comboBox (useful so we don't create duplicate items in list)
        self.dlg.comboBox.clear()
        # Add all items in list to comboBox
        self.dlg.comboBox.addItems(layer_list)
        # Clear comboBox_2
        self.dlg.comboBox_2.clear()
        # Add all items in list to comboBox_2
        self.dlg.comboBox_2.addItems(layer_list)

        def layer_field():
            # Identify selected layer by its index
            selectedLayerIndex = self.dlg.comboBox.currentIndex()
            selectedLayer = layers[selectedLayerIndex]
            # Identify fields of the selected layer
            fields = selectedLayer.pendingFields()
            # Get field names of the fields
            fieldnames = [field.name() for field in fields]
            # Clear comboBox_5
            self.dlg.comboBox_5.clear()
            # Add field names to comboBox_5
            self.dlg.comboBox_5.addItems(fieldnames)

        def table_field():
            # Same comments as above
            selectedLayerIndex = self.dlg.comboBox_2.currentIndex()
            selectedLayer = layers[selectedLayerIndex]
            fields = selectedLayer.pendingFields()
            fieldnames = [field.name() for field in fields]
            self.dlg.comboBox_3.clear()            
            self.dlg.comboBox_3.addItems(fieldnames)
            self.dlg.comboBox_4.clear()
            self.dlg.comboBox_4.addItems(fieldnames)

        # When changing layer in comboBox, run the function "layer_field()"
        # to refresh the field names in the associated comboBoxes
        self.dlg.comboBox.currentIndexChanged.connect(layer_field)
        self.dlg.comboBox_2.currentIndexChanged.connect(table_field)