Hi, I'm not the biggest expert, but here is something to start with:
Have you tested your Uno with any other code? Could be that the hardware isn't working right. Try loading this sketch first:
void setup()
{
Serial.begin( 9600 ); // Opens the serial line @ 9600 baud
delay( 10 );
Serial.println( "Hello, world" ); // prints out on the Serial monitor
}
void loop()
{
}
This should print "Hello World" on your serial monitor.
If this works, there is something weird about the RTC code you are trying to run. You should upload it, so we can take a look.
If the Hello World thing doesn't work, there is probably a big hardware problem.
This is just a starting point. Tell us what you find.