Does this code fall into a design pattern?
The initialise reminds me a factory but it's not exactly a factory. It's fine if you don't use the same if-elseif structure in your code elsewhere. If you do pull out it to a factory class.

Just some idea:

1, Consider using an enum instead of the integer typeCode (eliminate magic numbers).

2, Maybe you should throw an IllegalStateException in the end of the initialise method with a message "Invalid typecode".

3, The same is true for the setTypeCode() method. Or set explicitly typeCode to 0 in the last line of the method if 0 is a valid value. (This is the implicit default value of the private int typeCode field.)

4, I'd rename the setTypeCode() to calculateTypeCode() which would return a TypeCode enum. Now it's more or less temporal coupling.

5, You should create a new getNextScreen() method in the CustomDialog class with the first four line of the displayDialog() method. Then call the getNextScreen() from the displayDialog() methods. It would remove some code duplication.

// CustomDialog class code
public Dialog createNextScreen() {
    DialogType dialogType = new DialogType();
    DialogBuilder dialogBuilder = new DialogBuilder();
    AbstractDialogFactory abstractDialogFactory = 
        dialogType.getDialogType(DialogEnum.EXCLAMATION_DIALOG);
    Dialog nextScreen = dialogBuilder.buildDialog(abstractDialogFactory, text);
    return nextScreen;
}




// OkExclamationDialog, ExclamationDialog class code
public void displayDialog() {
    Dialog nextScreen = getNextScreen();
    ...
    ScreenController.displayNextScreenFadeTransition(nextScreen);
}