UNO board turned hot
This depends on what you mean by hot. If the atmega chip is so hot that you cannot touch it without burning yourself, then you are either supplying incorrect power to the chip or you are drawing too much current from the pins. This should not be the case. If it is, there could be something wrong with your arduino board, something wrong with your LCD board or hopefully it is wired incorrectly. I would check to make sure that the wires that are powering the backlight on the LCD board are connected to one of the power rails and not to an output pin of the arduino. 

If the arduino is kind of warm to the touch, I would not worry about this.
If the voltage regulator on the board is really HOT, you need to supply power to the LCD board from another source. 

If none of this fixes it, go through with a multimeter and measure current on each line going to the lcd board. I think the Arduino has around maximum of 5mA-10mA per io and 20mA to 100mA total for all pins. If any of the pins or the system as a whole exceed this, you need to use transistors and an external power source to drive the pins on the LCD board.